algorithmic and approximation in free constructions
TRANSCRIPT
Algorithmic and approximation problems in free constructions
by
Ekaterina Lioutikova
Department of Mathematics and Statistics McGill University, Montréal
March 1999
-4 thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree
of Doctor of Philosophy
@ Ekaterina Lioutikova, 1999
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Abstract
In the first part of this thesis, we introduce a claçs of groups whose m a i -
mal abelian subgroups are either malnormal or modules over a given ring .-\?
and use free constructions to descnbe explicitly the structure of tensor A-
completion for groups from this class. .As a corollary, we obtain a description
of tensor completion for groups of the form F/iV1, in particular. free solvable
groups. The tensor completion GQ of a torsion-free hyperbolic group over
the field of rational numbers can be similarly described in terms of free con-
structions. Using this description of @, in the second part of this thesis we
provide an algorithm that decides whether a finite system of equations over
G9 is solvable, and if it is, finds a solution. In the third part of this thesis we
study residual properties of Lyndon's group F ~ [ " I (the free exponentiiil group
over the ring of integral polynomials Z[z ] )? whose structure also involves free
constructions. We show that F ~ [ ' ) iç conjugately resiclually free, i.e.. it is
possible to map F"['] to n free group prese~ ing the nonconjiigncy of two
elements.
Résumé
Dans la première partie de cette thèse on introduit une classe des groupes
dont les sousgroupes maximaus abéliens sont conjugués-séparés ou des mod-
ules sur un anneau A, et on utilise des constructions libres pour décrire
explicitement la structure de la complétion il-tensorielle pour les groupes de
cette classe. Par conséquence, on obtient une description de la cornplétion
tensorielle des groupes du genre FIN ' , en particulière, des groupes résolubles
libres. La complétion tensorielle GQ d'un groupe hyperbolique sans torsion
sur Q peut être décrite d'une façon similaire à l'aide de constructions libres.
En utilisant cette description de GQ, dans la deuxième partie de cet te thèse
on construit un algorithme qui décide si un système fini d'équations sur GQ
soit résoluble, et si oui, trouve une solution. Dans la troisiérne partie de
cette thèse on étudie des propriétés résiduelles du groupe de Lyndon fZi'1
(le groupe exponentiel libre sur l'anneau des polynômes Z [ x ] ) . dont la struc-
ture emploie également des constructions libres. On montre que ~ ~ [ ~ i est
résiduellement libre B l'égard de la conjugaison, i.e., il est possible d'appliquer
F ~ [ ~ I dans un groupe libre si bien que les images de deus éléments noncon-
jugués sont nonconjugués.
Acknowledgement s
1 would like to take this opportunity to express my gratitude and appreci-
ilteion to Professor Kharlampovich, who, wit,h her ewr-r~ady acivicel critirisrn
and creative remarks, has made it a pleasure to have her as a supervisor. Her
enthusiasm and dedication to mathematics have been an excellent example
for me.
1 am very grateful to Professor blyasnikov for numerous helpful discus-
sions. His ideas and suggestions were of invaluable assistance in the prepa-
ration of Chapter 2 of this thesis.
1 would like to thank Yatural Science and Engineering Research Council of
Canada: Institut des Sciences 'Ylathématiques, and 3lcGill Major Fellowship
Foundation For their finanical support, without which this work would have
been impossible.
Special thanks to my husband Serguei for his support. patience. and
understanding, and to my son hndrei for being such a n inspiration.
Contents
1 Introduction 2
'1 1.1 Free constructions . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 1.2 Exy uiieiitial groups orid terisor corriplecions j
1.3 Equations over groups . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Lyndon's group . . . . . . . . . . . . . . . . . . . . . . . 12
1.5 Staternent of originality . . . . . . . . . . . . . . . . . . . . . . 14
1.6 Contribution of CO-authors . . . . . . . . . . . . . . . . . . . . 14
2 Tensor completion for a certain class of groups 15
2.1 Definition and properties of the claçs K.+ . . . . . . . . . . . . 16
. . . . . . . . 2.2 Tensor completion b r groups of the form F1.V'. 31
3 Equations iri the Q-completion of a torsion-free hyperbolic
VouP 38
3.1 Introduction and basic definitions . . . . . . . . . . . . . . . . 35
3.2 Some properties of the Cayley graph of H . . . . . . . . . . . . -1-1
3.3 Construction of canonical representatiws . . . . . . . . . . . . 51
3.4 Middles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.5 Shrinking of the middle t-strips . . . . . . . . . . . . . . . . . 61
4 Lyndon9s group is conjugately residually free 72 ri. 4.1 Prehminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . t3
4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5 Conclusion 105
1 Introduction
1.1 Free constructions
The notion of a free constructible group was first introduced by C'. Rerneslen-
nikov [59] in connection with the problem of description of finitely generated
groups acting freely on A-trees. -4ccording to his definition, a group is free
constructible if it can be obtained from a nonabelian free g o u p using a finite
number of "admissible elementary free constructions" , i.e., HNN-extensions
with certain restrictions on the associated subgroups. Since then. the no-
tion of a constructible group has expanded to include groups obtained from
free groups (as well as more complicated groups that have a lot in common
with free groups, such as torsion-free hyperbolic groups and, more gener-
ally, CS.4-groups) using a sequence of free constructions. i.e.. free produc ts.
free products with amalgamation, and HNN-extensions, provided that the
associated (amalgamsted) subgroups satisfy some natural conditions. Con-
structible groups have attracted a lot of interest in the recent years, since
many essential properties of groups, such as the property of being hyperbolic.
turn out to be preserved under free constructions of certain types.
In his book [li] hl. Gromov stated that if G1 and G2 are torsion-free
hyperbolic and Li and C' are maximal cyclic subgroups in Gi and G2 respec-
tively, then the amdgamated free product Gi *o=v G2 is also hyperbolic.
In [4] it was shown that an amalgamated product of two hyperbolic groups
with a cyclic subgroup amalgamated is automatic, and conditions were es-
tablished under which amalgamated products and HNX-extensions of finitely
generated free groups are asynchronously automatic. In [3? 121 it was proved
that if an amalgamated free product Gl *u G2 is automatic, then both Gl
and G2 are automatic provided that the amalgamated subgroup li is finite.
In [6] M. Bestvina and M. Feighn proved a combination theorem for neg-
atively curved spaces rvhich allows one to show that the fundamental groups
of certain graphs of hyperbolic groups are themselves hyperbolic. Their the-
orem implies, in particular, that the amalgamated product of two hyperbolic
groups with virtually cyclic amalgamated subgroups is hyperbolic if and only
if a t least one of the amalgamated subgroups is alniost malnormal. (Recall
that a subgroup !CI of a group G is called almost malnormal if for every
g E G \ M the intersection M n h i 9 is finite.) This result also follows from a
theorem by O. Kharlampovich and A. àlyasnikov [NI, where they show that
the anialgamated product Gi *u,=cr, G2, where GI and G2 are hyperbolic.
LIi 5 Gi is quasiisornetrically embedded, and UI almost malnormal in GI. is
again a hyperbolic group. 0. Kharlampovich and .A. Myasnikov also prove
in (2-11 that if G is a hyperbolic group, H = (G, t 1 Ut = V ) is a separated
HNN-extension (meaning that either Li or 1' is alrnost malnormal and the
set U n g-'Vg is finite for al1 g E G), and, in addition, the subgroups L; and
V are quasiisometrically embedded in G, then H is hyperbolic. .As a corol-
lary, it is shown that if G is a hyperbolic group and A and B are isomorphic
virtually cyclic subgroups, then the HNN-extension H = (G, t 1 At = B) is
hyperbolic if and only if i t is separated. This result, as well as the above-
mentioned criterion for the hyperbolicity of the amalgamated free product
of two hyperbolic groups with virtudly cyclic amalgamated subgroups, was
established independently by K. Mikhajlovskii and A. Ol'shanskii in (471.
The results of [24] imply a number of other corollaries: in particular. that
HNN-extensions (amalgamated products) of hyperbolic groups with finite
associated (amalgamated) subgroups are hyperbolic.
A group G is termed residually finite, if for eacli non-trivial g E G there is
a finite quotient G -t G such that the image of g in G is non-trivial. In 1987
M. Gromov posed an intriguing problem which to this day remains open: is
every hyperbolic group G residually finite? Although in general this problem
is very difficiilt, the answer is affirmative for some constructible groups G.
D. Wise proved in [64] that groups which split as certain graphs of virtually
free groups are residually finite. In particular, he çhowed that if G = -4 *,II B
is an amalgamated free product? where -4 and B are virtually free and .II
is a finitely generated almost malnormal subgroup of -4 and B. thcn G is
residually finite.
Free constructions have also proved to be iiseful in the investigation of
subgoup separability, which is a much stronger property than residual finite-
ness. -4 group G is called subgroup separable if every finitely generated sub-
group is the intersection of finite indes subgroups of G. For instance. free
groups [19] and surface groups [62] are subgroups separable. Brunner, Burns
and Solitar [8] proved that the free product of two free groups amalgamat-
ing a cyclic subgroup is subgroups separable. Using a topological approach,
hl. Tretkoff [63], R. Gitik [14] and G. Niblo [53] further generalized the results
of [8]. In [65], D. Wise investigated the conditions under which a group G
that splits as a finite g a p h of free groups with cyclic edge groups is subgroup
separable and showed that this is the case if and only if G is balanced (i.e.,
for any non-trivial element g E G, if gn is conjugate to gm, then n = +m.)
In (501, -4. Myasnikov and V. Rerneslennikov introduced the class of CS.&
groups, or groups with conjugate separated maximal abelian subgroups. Re-
cal1 that a subgroup AI of a group G is conjugate separated, or malnormal. if
M f M x = 1 for any x E G\.bl. The class of CSA-groups is quite wide: for ex-
ample, it contains free groups, torsion-free hyperbolic groups, groups acting
freely on A-trees, and universally free groups. In other words, the CS-A-class
comprises groups that are in some sense "close" to Free groups. Many results
that are valid for hyperbolic groups, or groups acting freely on A-trees. can be
generalized to this class. A. Myasnikov and V. Remeslennikov showed in [SOI
that for groups without elements of order 2 the CS.-\-property is preserved
under free products and amalgamated free products of specific type. called
abelian extensions of centralizers. In [13] D. Gildenhuys, O. Kharlampovich.
aiid A. bIyasnikov described nat ural condit ions under w hich HXN-extensions
and amalgarnated products of CS.-\-groups are again CS.!. They proved. in
particular, that a separated HXN-estension of a CSX8-group (CSA. without
elements of order 2) mith malnormal associated subgroups is a CS-Y-group.
and, similarly, that an amalgamated product of CS..\*-groups with a malnor-
mal amalgamated subgroup is again CS.\'.
1.2 Exponential groups and tensor complet ions
The axiomatic notion of an exponential group was introduced by R.C. Lyn-
don in [39] as a tool in his description of the set of al1 solutions of a given
equation in a finitely generated free group. An A-group after Lyndon is a
group that admits an action of the ring A satisfying a certain set of axioms.
In particular, if A is the ring of integers, the concept of an A-group reduces
to that of an ordinary group. For A = Z,, the integers modulo m, the A-
groups are groups in which al1 elements have order dividing m. For -4 the
field of rationals, the A-groups are groups in which "extraction of roots" is
always possible and unique; questions related to this were studied by B.H.
Neumann [52], A.I. Mal'cev [G], and A. Kontorovich [31]. G. Baumslag (21
considered, more generally, the case where rl consists of those rationals that
can be written with denominator a product of primes belonging to some pre-
scribed set. Taking a different approacho P. Hall [20] and also M. Lazard [33]
considered groups admitting exponents from a ring more general tlian the
ring of rational numbers.
In [50] A. hlyasnikov and V. Remeslennikov developed a new notion of'
an exponential group, adding one more asiorn to Lyndon's definition. This
improved notion is more convenient because it coincides exactly witti the
notion of a module over a ring in the abelian case, whereas abelian exponen-
tial groups after Lyndon constitute a far wider class. These two definitions
coincide in the case of free groups.
The continuing interest in the theory of exponential groups is motivated
by many factors. First, many natural classes of groups are A-groups; for
esample, unipotent groups over a field k of characteristic zero are k-groups,
pro-pgoups are exponential goups over the ring of p a d i c integers, etc.
Second, the notion of an A-group is a natural generalization of the notion
of a module over a ring to the category of non-commutative groups. Third,
one can consider for an arbitrary group G its largest ring of scalars A(G),
over which G is an A@)-group. This notion is an analog of the notion of a
centroid for rings and algebras (see [34] for details) and plays as important
role in the theory of exponential groups. Finally, as s h o w in [48], A-groups
are very helpful in the investigation of model-theoretic problerns for non-
commutative groups.
In [49] it was demonstrated thrrt the kcy rolc in thc study of c:iponential
groups is played by the construction of an A-tensor completion G;' of a group
G, where A is an arbitra. associative ring with identity. It is uniquely
determined by a natural universal property. Tensor completions have turned
out to be an extremely effective tool in the study of free .-l-groups (see [SOI).
groups acting freely on A-trees, where A is any ordered abelian group (511,
and in producing interesting new examples of hyperbolic groups [?A].
In the general case, it is difficult to give a constructive description for the
tensor completion of an arbitrary group. However. there have been successful
attempts to give such a description in some particular cases.
The first step in this direction was made by G. Baumslag [2]. nho intro-
duced the Q-completion FQ of a free group F, and, what is most important.
described FQ explicitly as the union of an ascending sequence of groups, each
of which is a free product with a single amalgamation. This allowed him to
establish some properties of the group FQ. In [50] A. Myasnikov and C'.
Remeslennikov developed this approach, applying it to an arbitrary ring .4
of characteristic O and to the class of CSA-goups.
In [50] for an arbitrary CS-4'-group G the construction of tensor comple-
tion was obtained as an iterated tree extension of centralizers of the group
G. This result gives us a nice constructive description of the tensor comple-
tion for a very wide class of groups; however, it does not apply to groups
containing normal abelian subgroups (such as solvable groups). In chap-
ter 2 for a given ring A we introduce a new class which consists of
groups whose maximal abelian subgroups are either conjugate separated or
-4-modules (it contains, in particular, the class of CSA-groups.) The main
result of chapter 2 is an explicit description of the structure of tensor corn-
pietion for &-groups over an arbitrary ring of characteristic 0, which uses
free constructions of a specific type. As a corollary, we obtain a description
of tensor completion for groups of the form FIN', where F is a free group, :V
a normal subgroup of F and Nt = [N, NI, under certain natural restrictions
on the group F I N . In particular, this result applies to free solvable groups.
An interesting c o r o l l a ~ is that the tensor completion of a free nonabelian
solvable group is, in general, not solvable.
In the study of tensor completions the case where a group G is a sub-
group of its A-completion (Le. G is A-faithful) is of particular interest. In
[SOI it was proved that the class Fa4 of al1 .A-faithful groups is universally
axiomatizable and closed under subgroups. direct products and direct lirnits.
and, in the case where .A is an integral domain of characteristic O , contains al1
residually torsion-free nilpotent groups. In chapter 2 we prove that the class
of torsion-free faithful cl-groups over an arbitrary ring -4 of characteristic
O is closed under free products. Moreover, the construction of the tensor
cornpletion for free solvable groups implies that every free solvable g o u p is
faithful over any torsion-free ring A.
1.3 Equations over groups
The theory of exponential groups is one of the important developments in
group theory that was inspired by the famous and extrernely challenging
problem raised by A. Tarski around 1945. The problem consists of two
parts: (1) is the elementary theory of a free group decidable? and (2) do
the elementary theories of all nonabelian free groups coincide*? 0. Khar-
lampovich and .4. Myasnikov [28] recently announced a theorem that gives
an affirmative answer to Tarski's questions. They show that the free group
F ( a l , . . . , a,) freely generated by a l , . . . , a, is an elementary subgroup of
F ( q , . . . , an, . . . , a,,,) for every n 1 2 and p 2 0; moreover, the elementary
theory T h ( F ) of a free group F is decidable, even if the language contains
constants from F.
During the investigation of Tarski's problem, a particular case about the
decidability of the positive 3-fragment of the elernentary theory of a free
group, and about the construction of sets defined by positive 3-formulas.
turned out to be of great interest. These questions are equivalent to the
question about the algorithmic solvability of a system of equations over a free
group, and about the description of the set of al1 solutions of such systerns.
It is not surprising, therefore, that systems of equations over groups have
received a lot of attention and have become one of the main streams of corn-
binatorial g o u p theory. The first general results in these area are due to
R. Lyndon [38], who constructed an algorithm for solving equations in one
variable and described the solution sets of such equations using parametric
words. A. Lorents [36] noticed that Lyndon's algorithm can also be applied
to systems of equations in one variable. In 1968 Lorents [37] and indepen-
dently K. Appel [1] described the solution set of an arbitra- equation in one
variable over a free group. The algorithmic decidability for systems in which
every equations contains at most two variables was establislied, under some
resrrictions, by Yu. Khrrielevskii [29, 301 wid iu general case by Yu. Ozhigov
[ 5 51.
The work of A. Mal'cev, who described in 1966 the solution set of the
equation [x, y] = [a, b] over the free group with generators a, b, lies in the
origin of the study of solution sets of quadratic equations over free groups,
i.e.! equations in which every variable occurs exactly twice. L. Comerford
and C. Edmunds [9, 101 and R. Grigorchuk and P. Kurchanov [lj. 161 de-
scribed the solut ion sets of standard quadratic equat ions oyer arbit rary free
groups. This completes the quadratic case. because it follows froni the work
of A. Hoare. -4. Karras and D. Solitar [-1? 221 that every quadratic equation
is automorphically equivalent to a standard one.
In 1952 G. Makanin [43] proved the crucial result about the algorithmic
decidability of the Diophantine problem over a free group. He proved that if
a given equation over a free group F haç a solution in F, then this equation
has a solution of bounded length, and this bound can be effectively computed
from the equation itself. In his paper Makanin developed a powerful tech-
nique to deal with equations over free goups, as well as over free semigroups.
Makanin's work made it possible for A. Razborov [56, 571 to describe the so-
lution set of an arbitrary system of equations over F. Shortly aftenvards,
G. Makanin (411 extended these results to prove that the universal theory of
a free group F, i.e., the set of al1 univenai sentences that are true in F , is
decidable.
Since the question of the algorithmic decidability of systems of equations
is a generalization of the word and conjugacy problems. it mises naturally not
only for free groups but also for other, more complicated. classes of groups.
E. Rips and 2. Sela 1611 developed an effective procedure to decide wliether
a system of equations in a torsion-free hyperbolic group has a solution: they
used canonical representatives in hyperbolic groups to reduce a system of
equations in a torsion-free hyperbolic group to a finite set of systems in a
free group. Note that the case of qudrat ic equations in hyperbolic groups
was solved earlier by 1. Lysenok 1.111. O. Kharlampovich and A. blyasnikov
[X] constructed an aigorithm to solve finite systems of equations in a free
Q-group FQ: and, more generally, in a free Q,-group, ahere n is a recursive
set of primes.
In chapter 3 of this thesis we establish the algorithmic solvability of fi-
nite systems of equations over the Q-completion of a torsion-free hyperboiic
group. If G is torsion-free hyperbolic, GQ is a constructible group, i.e., it
can be described as a union of an effective chain of hyperbolic subgroups
obtained from G by means of free constructions of a certain type. Based on
this description of Ga, we show the existerice of an algorithm that decides
if a given finite system of equations over GQ is solvable. and if it is. Ends a
solution.
1.4 Lyndon's group ~ ~ [ ~ l
R. Lyndon's study of solution sets of equations over a free group leci him to
the notion of a group with parametric exponents in an associative ring -4.
In particular, he described and studied the free exponential group F ~ [ ~ ] over
the ring of integral polynomials Z[z]. One of the crucial results of his study
was that the group F~(*] is discriminated by F, i.e., for any finitely ma-
non-trivial elements in F ' [ ~ ] there exists a homoniorphism 4 : F ~ [ ~ ] + F ?
which is the identity on F, such that the images of the given elements under
# are also non-trivial. In 1989 V. Remeslennikov [5S] established a surprising
connection between residual properties of groups and their universal theories,
namely, that a finitely generated group H c m be discriminated by a non-
abelian free group F if and only if H has the same univenal theory as F. This
further emphasized the role of F'['] in the investigation of Th(F) . A modern
treatment of ~ ~ i ~ ] was given by A. blyasnikov and Y. Remeslennikov in [SOI.
They showed, in particular, that F ~ [ ' I can be obtained from F by npplying
infinitely many free constructions of a specific type, called free estensions
of centralizers. in the same paper [SOI? Myasnikov and Remeslennikov con-
jectured that finitely generated group is discriminated by a nonabelian free
group F if and only if it is embeddable into F ~ ~ I . This conjecture was solved
affirmatively by O. Kharlampovich and A. hlyasnikov in a series of papers
[26] and [27]. In this way, a description of finitely generated groups which are
discriminated by F as well as a description of al1 finitely generated groups
that have the same universal theory as F was obtained. Moreover, in [27],
the structure of finitely generated subgroups of Lyndon's group is described.
It turns out that every finitely generated subgroup of F ~ [ ~ I (and hence, every
finitely generated group discriminated by a free group) can be obtained frorn
a free c o u p of finite rank by finitely many free products, free extensions of
centralizers, free products wit h amalgamat ions and HNX-extensions? where
amalgamated jassociated j subgroups are free abeiian of finite rank. This irn-
plies, in particular, that every finitely generated fully residually free group is
finitely presented. Lyndon's group and its subgroups play a vital role in the
technique employed by O. Kharlampovich and A. Xlyasnikov when dealing
with the elernentary theory of a free group (see (281).
In chapter 4 of this thesis, we show that Lyndon's group is conjugately
residually free, i.e., it is possible to map F ~ [ ~ ] to the free group F preserving
the nonconjugacy of two elenients. Some residual properties of F ~ [ ~ ] relative
to conjugacy were studied by J. Bridson in [ T l , where it was shown that the
conjugacy problem in ~ ~ t ~ ] is solvableo and the property of being conjugately
residually free was established for a single free extension of centralizer of a
free group F.
-4lthough the fact that ~ ~ 1 ~ 1 can be approximated by F with respect to
conjugacy is interesting in its own right (consider, for example, the results of
L. Ribes, D. Segal and P. Zalesskii [60] on conjugacy separability of certain
types of free products with amalgamation), it is also important in connection
with the problem of solving equations over ~ ~ [ ~ 1 and, in particular, with the
following question. Suppose that f (1 ,6 ) = 1 is an equation in variables f
with coefficients C in such that for every homornorphism # : F ~ I ~ ~ + F
the equation f (3, F') = 1 has a solution in F. 1s it then true that f = 1 has
a solution in Lyndon's goup? Although in general this is not quite true (and
we will formulate a more precise conjecture in the conclusion), the question
has an affirmative answer for some common simple equations, such as the
equation x-'clx = CZ.
1.5 S tatement of originality
The material presented in Chapters 2, 3, and 4 of this thesis is new and con-
stitutes original scholarship in niathematics. Some auxiliary results? included
in this t hesis to make it reasonably self-contained, are clearly ident ified as
previously known, and the reader is referred to the original sources.
1.6 Contribution of CO-aut hors
Chapter 3 of this thesis is baçed on the author's joint paper mith 0. Kharlam-
povich and A. Myasnikov [23]. In their earlier paper [25 ] , 0. Kharlampo~ich
and -1. Pvlyasnikov established the solvability of finite systems of equations
over a free Q-group F? Prof. Kharlamporich brought this paper to my at-
tention and pointed out that some of the essential results in [?5] apply. more
generally, to the case where, instead of F' a torsion-free h:vpperbolic group
is considered. Under the guidance of Prof. Kharlampovich, I modified the
rnethods described in [25], as well as developed additional rnethods specific
for the case of hyperbolic groups, to obtain the solvability of systems of equa-
tions over the Q-completion of a torsion-free hyperbolic group, and prepared
a paper [23] for publication. Some of the ideas and results first introduced by
O. Kharlampovich and A. hlyasnikov in (251 are vital to the case 1 considered
and thus constitute an integral part of [23]; whenever such results are used
in Chapter 3 of this thesis, the source is clearly acknowledged.
2 Tensor completion for a certain class of groups
The notion of a tensor completion plays a key role in the study of exponential
groups. For an associative ring A with identity and a partial .4-group G. the
tensor A-completion Ge4 is an A-group defined in terms of a natural universal
property. Since the definition is far frorn being constructive, an important
problem is to describe the explicit structure of the teiisor completion of a
group. There have been successful attempts to provide such a description
in some particular cases. In 1960, G. Baumslag (21 constructed FQ as a
union of an ascending chain of groups each of which is a free product with
amalgarnation. In 1993, A. Myasnikov and V. Remeslennikov [SOI developed
a similar approach to describe the tensor cornpletion of CS.4-groups (i.e..
groups whose maximal abelian subgroups are malnormal) over an arbitrary
torsion-free ring A. Their result applies to a wide class of groups. including
free, torsion-free hyperbolic groups, groups acting freely on A-trees. etc.; it
does not apply, however, to groups containing normal abelian subgroups.
such as solvable groups.
In this chapter for an arbitrary associative ring .A with identity Ive in-
troduce a new c lu s of groups &, which contains, in particular, the class of
CSA-groups, and investigate its properties. The main result of this chapter
is an explicit description of the structure of tensor completion for ICA-groups
over an arbitrary ring of characteristic O in tenns of free constructions. As a
corollary, we obtain a description of tensor completion for groups of the form
FIN', where F is a free group, N a normal subgroup of F and iV' = [ N , NI,
under certain natural restrictions on the group F I N . In particulnr, this re-
sult applies to free solvable goups. An interesting corollary is that the tensor
completion of a free nonabelian solvable group is, in general, not solvable.
In the study of tensor completions the case where a group G is a subgroup
of its A-completion (i.e. G is -4-faithful) is of particular interest. LVe prove
that the clriss of torsion-frcc hithful -4-groups orer an arbitrnry ring -4 of
characteristic O is closed under free products. Moreover, the construction of
the tensor completion for free solvable groups implies that every free solvable
group is faithful over any torsion-free ring il.
This chapter is based on the author's published paper [35].
2.1 Definition and properties of the class KA.
First of al1 let us briefly recall some basic notions, which will be enough for
independent understanding of t his cliapter. These notions are discussed in
detail in (491 and (501.
Let -4 be an arbi t rav associative ring with identitj* and G a group. Fis
an action of the ring A on Gy Le. a map G x -4 -t G. The result of the action
of a E A on g E G is written as ga. Consider the following axioms:
4. [g, h] = 1 (gh)" = p h a .
Definition 2.1 A grovp G zs called an -4-exponential group (or simply an
A-group) zf there is an action of A on G satzsfying axions 1)-4).
Definition 2.2 A grovp G 2s a partial -4-group i f gQ zs defined for some
g E G, o E A, and aPoms 1)-4) hold wherever the action is defined.
One c m also define the notions of an A-homomorphism and a partial
A-homomorphism in a natural way.
The basic operation in the class of partial A-groups is the operation of
tensor cornpletion, which is defined as follows.
Definition 2.3 Let G be a partial -\-group. Then an -4-group G" is called
a tensor completion of G , zf
1. there ezists a partial .I-hornomorphism X : G + Ge-' such thut X(G)
generates as an .A -group;
2. for eue? A-group H and e v e q partial A-homomorphism y : G + H there exists an A-homomorphism 0 : Ga4 + H svch that 9 = O A:
Definition 2.4 A partial &group G is called fuithful over A, if the canonicaf
homomorphisrn X : G GA zs injective.
Now let us introduce a new class of groups which will be the focus of
our attention throughout this chapter. Recall that a subgroup H of a group
G is said to be conjugute sepamted (or m a l n o m a l ) if H n Hz = 1 for al1
X E G \ H .
Definition 2.5 A partial -4-group G belongs t o the class &, if for e u e G y
maximal abelian subgroup iCI O/ G either M zs conjugate separated o r M is
a n A-module (i. e., there is a n action of -4 o n M .)
In the following proposition we establish some basic properties of K..\-
groups.
Proposition 2.1 Let G E ICA.
1. Suppose that .CIl # LI- are maximal abelian subgroups of G , and ut
leust one of XIi 2s conjugate separated. Then Mi n dl2 = 1.
2. If AI 5 G is a maximal abelian subgroup of G which is conjugate sep-
arated, then AI coincides with the centralizer 01 any of its nontrivial
elements.
1. Assume that !\Il is conjugate separated. Suppose 1 # x E JIl n M2. Take
y E iI12 \ MI. Then x = y-lxy is both in Ml and ICI,Y, but ilII is conjugate
separated. Therefore hl2 c Ml, and since both are maximal abelian, MI =
A& - a contradiction.
2. Let 1 # x E iV, suppose that [ x , g ] = 1 and 9 fl M. Then we have
x = y-Lxy, so that M n MY # 1 - a contradiction. a
In our description of the tensor completion for groupç from the class K A ,
we will frequently use the following constructions.
Definition 2.6 Let G be a group, Cc(u) = C the centralizer of an elernent c
from G, and # : C -t H u rnonomorphzsm of groilps such that d(o) E Z ( H ) .
Then the group
G ( u , H ) = ( G * H ( C = C m )
i s called an eztension of the centralizer C&) b p the group H with respect to
4. An extension of a centralizer is called abelzun if H is an abelian group.
The next construction allows one to extend a set of centralizers at the
same tirne.
Definition 2.7 Let C = {Ci, = & ( u t ) 1 i E 1} be a set of centralizers in a
group G. Suppose di : C, Hi i s an ernbedding of C, into H, such that
The fundamental group of this graph zs called a tree extension of centralizers
from C and is denoted by G(C, 'H, a), where
By definition, the group G(C, 31, @) is the union (direct liniit) of the
chain of groups
rvhere the set of centralizcrs C = {Ca 1 a < A } is ;vcll ordcrcd. C =
Go, Ga+i = Go(vaii< da : Ga 3 Ga+! is the canonical embed-
ding, and G, = lim, Ga (here cu < A/ ) for a limit ordinal 7.
Before stating the main theorern, let us describe the construction of com-
plete tensor extension of centralizers of a group G by a ring d (this termi-
nology was introduced in [50]). Here we will assunie that G is a nonabelian
group from ICA. and al1 maximal abelian subgroups of G are faithful over the
ring A.
Let M = {!\fi 1 i E I } be the set of al1 maximal abelian subgroups of G
such that:
1. for any M E M M is not an .A-module:
2. M contains one and only one representative of every conjiigacy class
of maximal abelian subgoups of G that are not -4-niodules.
Consider the tree extension of centralizers G* = G ( M , Hl a), where
U = {hIi @ -4 1 i E 1), and is the set of canonical embeddings
$i : iC[i ~f Mi @ A. Each $i is an embedding since Mi are faithful over -4 by
assump t h .
Iterating this construction up to level u, we obtain the chain of goups
C; = @O) < ~ ( 1 ) < . . . < @") < . . ., = ( ~ ( n ) ) ' .
Definition 2.8 The group G @ -4 = Un,, G(") zs called the complete tensor
extension of centmlzzers of G.
Main Theorem. Let A be a ring of characteristic 0, and G a torsion-free
group from the class K.+ If all maximal abelian subgroups of G are fuithful
over A, then the tensor cornpletion G.4 zs the complete tensor extension of
centralirers of G by the ràng .A.
For the proof of the Main Theorem, we will need the following results
about K;l-groups. Denote by ICl the class of al1 K;l-groups without elements
of order 2.
Theorem 2.1 The class Ki is closed under extensions of centra1zze.r~ by
abelian groups without elements of order 2.
Proof Let G E Ki, AI a conjugate separated maximal abelian subgroup of
G, and B an abelian subgroup without elements of order 2 such that there
esists an embedding 4 : J I c+ B. The extension of M by B iç defined by
G(M, B) = G * M B. Observe that G ( M , B) has no elernents of order 2 (see
(42: cor. 4.451).
The following lemma describes maximal abelian subgroups in G(M, B) .
Lemma 2.1 If N 29 a maximal abelian subgroup in G(M, B ) , then one of
the following is true:
1. N 5 G~ fo r some h E G(il1, B) (more precisely, N = for s o m e
mazirnal abelian !VI < G);
2- IV = Bh for some h E G(M, B) ;
3. N = (z) with IIzII 1 2 , whwe llrll stands for the cgclically reduced
length of 2 .
Proof Let r E N, r $ 1. Thi r i are th rw possihilities:
1. x E Mh for some h;
2. r E Gh or BA (but not in Adh);
Suppose that y E iV, so that [x, y] = 1.
1. x E MlL. Conjugating by h if necessary, we may assume that x E h l . Let
y = m g l bl . . . g,b, be the canonical forrn of y , where m E JI. gi are right
representatives of G modulo hl, bi are right representatives of B modulo d l ,
and g, # 1. Then xy = yz implies
so that g,, and g,,x are in the same conjugacy class of G by 31, and we have:
g,x = zig, for some xi E M. Hence d P n d l # 1, but M is conjugate
separsted - a contradiction. It follows that y = mbl E B, wtiich is case 2
of the lemma.
2. x E Ch (resp. Bh), then by the description of commuting elements in a
free
and
product with amalgamation (see [50] or [42]), y is in Gh (resp. in Bh).
we obtain case 1 or 2 of the lemma.
3. Finally, suppose that llxll 2 2 and [x,y] = 1. Then x = g-'rnlgzn,
y = g-Lmzgzm, mi E Ad, llrll 2 2' and z, g-lrnlg and g - h g commute
in pairs. Conjugating by g-'y we reduce the problem to the case where
x = mlzr, y = m2z;", zl = gsg-', and z l cornmutes with ml and m2.
If ml # 1 or rn? # 1. then zl E B according to the proof above. but
this contradicts the condition I(zII 1 2. So ml = 1 = rn?, and we have
rn x = zy, y = Zi .
Now let t be another element of N. The same argument shows that
I = $, 2 = Z! for some 22 with 1 1 ~ 1 1 2 3. Then al1 elernents of the subgroup
H = ( z l , z 2 ) have length 2 2. Indeed, let u E H and llvjl = 1. Observe that
2 = zf = r t is in the center of H. So [u. :y] = 1, and the description of
commuting elements in a free prodiict with amalgamation implies that r ; is
in the same conjiigate to one of the factors as u. This is impossible. since
llzr11 1 2. This allows us to apply the following description of subgroups
of a free product with amalgamation Gl *+FI G2 : if Qx H n G: = 1. then
H is a free group (see (42. p. 243, cor.4.9.2]). It easily follows that H is
cyclic. siiice it has a nontrivial center. We may assume that q and q are
not proper powers, therefore 21 = z?. Hence !V is cyclic, and ive have case 3
of the lemma. C l
Let us continue the proof of the theorem.
Let M = NF be a masimal abelian subgroup of G ( M , B) of type 1. We
need to prove that either iVx n A: # l implies x E N , or N is an A-module.
Conjugating by hdl if necessary, we may assume that N = N1 < G. Suppose
that there exist nontrivial elements f , g E N such that f* = g. Then.
according to (50, Lemma 21, f Y = g for some y E G. But then N n .VY # 1.
Since N is maximal abelian in C, we have that either 1V is an -4-module
or N is conjugate separated. In the latter case' y E Y, so f = g and
[x, f ] = 1. Moreover, according to the description of commuting elements in
Irw products witli ariialganiatiou ( j i 2 i j, x E G (observe thac r is nor in the
amalgamated subgroup AI; indeed. M is not maximal abelian in G(M, -4).
so hl # N, and JI n JI = 1). Since :V is the centralizer in C of any of irs
elements, in particular f , we obtain that x E A'.
Now let iV = BY, g E G(i\l, B). Again? it suffices to consider the case
N = B. The result immediately follows if B is an .4-module. which is in
fact the only case we will be interested in. In general. it is not difficult to
prove that B is conjugate separated in G(M, B) . Suppose that B n Bx # 1.
x E G(M. B). Then f' = h for some f! h E B, and by (50. Lernma '21. f
and h are conjugate in B. But B is abelian. so f = h. Then jC = h iniplies
[ f ~ ] = 1, and since f is in B, so is 2.
The proof for the case that iV is infinite cyclic is covered by the following
two Lemmas.
Lemma 2.2 In cr free product of two groups &th amalgumation. the central-
izer of any element z such that ( 1 ~ 1 1 2 2 is cyclk.
Proof Let x E C(z), so that [r , x] = 1. Frorn the description of commuting
elements in a free product with amalgamation uve obtain that r = rn(un,
s = rnqum, where rnY,rna and u cornmute pairwise. -4s before, it easily
follows that mi = m2 = 1, so that z = un and x = um. Moreover, llull 2 2.
I t remains to show that every element of C(z) is a power of u. Let y E C(z):
then [y, z] = 1, and the previous argument shows that z = v k , y = v' for
some u with llull 2 2. Consider the group H = (u, v) . Using the fact that
un = vk E Z ( H ) and the same argument as in the proof of Lemma 2.1 (part
3), we obtain that H is cyclic. We may assume that .u and c are not proper
powers, so u = II.
Lemma 2.3 Any mm'mal abelian subgroup o ~ G * ~ B of t h e f o n n ( z ) , 1 1 ~ 1 1 2 2, zs conjugate separated.
Proof Let x - l r m x = zn for some m and n. Then llzrnll = IIznII, but since
llrnll = I n 1 .11z11, it follows that either m = n or rn = -n. If x - l r m x = zm,
then [x, zm] = 1, and hence P belongs to the centralizer of 2". By the
previous lemma, C ( z m ) is cyclic; it obviously contains z , so [x, ,-] = 1, but
( z ) is maximal abelian. Therefore x E (z).
If X-':"X = z - ~ , then [x2, zm]= l . It is crucial that G +,II B has no
elements of order 2, so that +* is nontrivial. We obtain that r2 E C(tm).
which is cyclic, and hence [x2, z] = 1. Furthermore. x2 is not in a conjugate
of a factor (otherwise z would be in the same conjugate, which is impossible
since 11 zll 2 2). Therefore ( 1 ~ ~ 1 1 2 2, and its centralizer is cyclic. Since C ( x 2 )
contains both x and r , [x, z] = 1, but (2) is masimal abelian, so z E ( z ) . O
This completes the proof of Theorem 2.1.
Theorern 2.2 The class ICA is closed under direct lzmits, in the case where
the canonical mappzngs are ernbeddinys.
Proof Let G = l im, Gi, il1 a maximal abelian subgroup in G which is not
an A-module. Observe that M = lim, Mi, where 1% = M n Gi.
Claim. içI, is a maximal abelian subgroup in Gis
Proof. Suppose that Mi is not maximal abelian in Gi, then there exists
a maximal abelian iq in Gi such that Mi < Ad:. Starting from 1LI:, we
construct a maximal abelian subgroup M' of G in the following way: embed
Ml into Ciil, denote by i\i;+, a maximal abelian subgroup of G,+I containing
Ml, and so on. At limit steps take unions. In G dl' # I I (çince .\.I n G, =
Mi # 1bC = M1nGi), and bot h are maximal abelian. so they are not contained
in each other. There are rn' E i W and m E il1 such that [m', ml # 1. Observe
also that since i\I is not an -4-module, there is a number j such that for al1
k > j iCIk is not an .+-module. So taking k sufficiently large, we may assume
that rn', rn E Gk and A& is not an -4-module. 8 y construction, .\ItnGk = .\IL
is a maximal abelian subgroup in Gk. The group hlk is contained in some
maximal abelian Nk < GI; Observe that !CI' # iVk (m is in :Vk but not in
Mi, and m l is in iC1; but not in Nk). and ML n iVk # 1 (both contain the
image of AI i ) . We may assume tliat k is large enough so that ,Vk is not an
.&module. (Indeed, if there is a j such that for al1 k. > j Nk is an -4-module.
it follows that lim, Nk = N is an -4-module. Since X obviously contains
M, and 1LI is maximal abelian in G, we have AI = !Y. But -11 is not an
A-module by assumption.) Since Gk is in class hl'' n !Vk # 1 means that
both are .&modules - a contradiction.
To complete the proof of the theorem, suppose that M < G is not an
A-module and Mx n M # 1. Then rn? = rn* for some ml, rn? E ilf, z E G.
There exists a k such that rnl,m2,x E Gk, so iCl,'nMk # 1. We assume that
k is large enough so that hlk is not an .A-module, and since Gk E I C A , A l k is
conjugate separated - a contradiction. Cl
Theorem 2.3 The class ECl is closed under tree abelian extensions of cen-
tralizers and iterated tree abelian extensions of centralizers 63 abelian groups
udhout elements of order 2.
Proof By definition, a tree (iterated tree) extension of centralizers of a
group G is constructed by transfinite induction on ordinals. Following this
construction, it suffiçes to prove by induction that every term of the chain
G = Go < Gi < . - * < Ga < - + - belongs to the class iC1 if the initial group
G is in IC:. For a nonlimit ordinal this was proved in Theorem 2.1. and for
a limit ordinal in Theorem 2 2 . [3
Corollary 2.1 If G E K:, und the ring .4 hus no udditiue elerrcrrits O/ order.
2, then the tree extension of centralizers G' is a K; -group.
Proof As proved in (50, Proposition 131, none of the groups AI, 8 .4 from
the construction of G* contains elements of order 2. The rest of the proof
follows from Theorem 2.3. 0
Lemma 2.4 The tree extension of centralizers G* is a partial A-group, and
the action of A on G is defined in G*.
Proof Let g E G, then there is a maximal abelian subgroup M of G
containing g. If M is an A-module, then the action of A on g is defined in
G, moreover in G*. Suppose that M is not an A-module, then 111 = ie for some i\fi € MM, and g = g: for some gi E hlii. 11n G* iCli v hli 8 -4,
so the action of A is defined on gi. For a E A, we let gQ = (gr)=. Since
G is a KA-group, M is conjugate separated. and we can extend this action
correctly to al1 conjugates of il1 to satisfy aviorn 3 from the definition of an
A-group. In this way, the action of A is well-defined in G' on al1 elernents of
G, since any two maximal abelian subgoups of G that are not -4-modules
intersect trivially. Cl
In what follows it is essential that G is torsion-free. and -4 is of charac-
teristic O.
Lemma 2.5 Al1 maxzmal abelion subgroups of G* are fuithful over -4.
Proof Each maximal abelian subgroup in G* is either an -4-module. or
infinite cyclic. In both cases, it is faithful over -4. In fact? for an infinite
cyclic group (2) we have:
Lemma 2.6 G* has the following universal propertp with respect to the CU-
nonical embedding i : G * G*: for any A-group H and for a n y purtiul
A-homomorphism f : G -t H there exists a partial A-hornomorphism
f * : G* -t H such that f = f ' o i .
Proof Let 1b.I be a maximal abelian subgroup of G belonging to M. Then
f ( M ) generates an abelian A-subgroup N in H. Using the universal property
of the tensor cornpletion for abelian goups, we can find a homomorphism
qnr : M A --+ N such that f = @l\,.l o &, where in[ : iCI v 111 8 A is the
canonical embedding. By the univeaal property of G* similar to that of a
free product with amalgamation (see [SOI), there exists f' : G* + H such
that f = f ' o i . O
Now Ive are ready to complete the proof of Main Theorem.
Let Ga = G, G("+') = ( G ( ~ ) ) * ~ where G* is dpfinpd as a h o v ~ . Thpn
G 8 -4 = Un,, G(") is the tensor rl-completion of G. Indeed. G 8 .4 is an
-4-group: if x E G @I A, then z E G(") for some n, and by lemrna 2.4, the
action of A on x is defined in GWCL).
It remains to check the universal property of the tensor cornpletion. Let
H be an -4-group and f : G + H a hornomorphism. For any n , / estends to
a homomorphism jn : G(R) + H of partial -4-groups. Then f4 = Un,, Jn is
an ri-homomorphism such that f = f 'oi, mhere i is the canonical embedding
i : G + G @ A . Cl
Corollary 2.2 Let G be o torsion-jree K.4-group. and -4 a ring of charac-
teristic O. Then G is fuithful ouer .4 if and onlg if e v e q abelian subyroup of
it zs faithful over A.
Theorem 2.4 Let -4 be a ring of chamcteBstic 0. Then the class FV4 of
faith ful to rsion-free -4 -groups is closed under free products.
For the proof of the theorem we will need the following
Lemma 2.7 Let Gi and G2 be groups withovt elernents of order 2. Then
in the free pmduct Gi * G2, any maximal abelzan subgroup of the fonn (2):
where llzll 2 2, is conjugute separated.
Proof First let us prove that the centralizer of z in GI * G2 is exactly ( z ) .
Let x E C ( r ) , then [x, z ] = 1. Consider the group H = (z, z). Clearly, H is
abelian. At the same time, H contains ( r ) , and since (z) is maximal abelian,
we have H = ( z ) . Therefore C ( z ) < (2). The inverse inclusion is trivial.
To prove that (z) is conjugate separated, suppose that (2)' n ( z ) # 1.
then x-lzmx = in for sorne m, n. Using the length argument as before. we
obtain that 1 m I=I n 1. There are the following two cases:
a) If m = n, we have I - ' ~ ~ X = zm, which is equivalent to [z, zm] = 1. The
group H = (x, 2) lias s nontrivial center (at l e m , it contains 2"). By the
Kurosh subgroup theorem ([42]). since H is a subgroup of a free product. it
is of the form H = F * (*,Ha). but it has a nontrivial center. and therefore
coincides with one of the factors. Since 2 E H and IIzII 2 2. H # Ha.
Hence H = F, so it has to be cyclic. Therefore [r, z ] = 1. whicli implies
x € C(z) = (2).
b) 1f m = -n, then x-'zmx2 = - vrn ? aiid by the same argument as above.
x' E (2). By [42. cor. 4.4.51, Cl *G2 contains no elements of order 2, so r2 + 1. Consider the group ( 2 , x) . Its center contains x2 and therefore is nontrivial.
In the same way as in a) we obtain that (z, x ) is cyclic. Conseqiiently, in this
case x E (2) too. 0
Proof of Theorem 2.4 Let G1, G2 E f i . Then GI ~t Gf and G2 v G:,
so GI *G1 ~t Gf *G;L.
Ciaim 1. Gf * Gt E KA.
Proof. Let hl be a maximal abelian subgroup of Cf * G$ By the Kurosh
subgroup theorem, we have the following two cases:
1. i\.I = (2) with llzll 2 2;
2. il1 < (Gf)= or M < (Gf) i , 2, y E Gf * G,A.
In the first case Ad is conjugate separated by lemma 2.7, and in the second
casc it is an -4-module (indeed, J I < JI" < GL, but since .W4 is an abclinn
group and M is maximal in Gf, ;CI = MA).
Claim 2. Any maximal abelian subgroup !id of Gf * G$ is fai thful over -4.
Proof. r ç l is either an A-module or infinite cyclic; in both cases it is
faithful over -4.
By the Main Theorem, G l * Gt is faithful over A. and hence it is embed-
dable into some A-group K:
This proves that Gi * G2 is faithful over -4. 0
2.2 Tensor cornpletion for groups of the form FIN' .
Let F be a free group, iV a normal subgroup of F, and NI = [N, .VI. In what
follows we will extensively use the theorem quoted below.
Theorem 2.5 Let FIN be torsion-free. Then two elements of FIN' corn-
mute if and only zf they either belong to rV/W or belong to the same cyclic
subgroup of F/.Wt.
Proof See [46] or (181.
Now consider the following conditions on a g o u p G:
1. G is torsion-free;
2. G is an R-group (terminology due to Kontorovich. see [XI): for
x: 9 E G xm = y" implies x = y; in other words, G is a group in which
roots that exist are unique;
3. for every element g of G there is a primitive element f such that every
root of g is a power of f . (Recall that f is called primitive if it is not
a proper power.)
Lemma 2.8 If FIN is un R -group, then so is FIN'.
Proof See (461.
Lemma 2.9 Let F/!V be torsion-free. If F1.V satisfies condition 3, then su
dues FIN'.
Proof Suppose for contradiction that for some g.V E F/.V there is no
primitive element with the required property. Then we can builcl an infinite
chain of the following form:
(mod N t ) , . . . g = g; (mod N t ) , gi = g,"' (rnod N I ) , . . . , gk-, = g,
where nk # 1 for dl k. Since Nt 5 N, the same equalities hold modulo N,
which contradicts to the fact that FIN satisfies condition 3. a
Corollary 2.3 Conditions 1-3 hold for free solvable groups.
Proof Condition 1 is trivial. Conditions 2 and 3 ob~iously hold for free
abelian (one-step solvable) groups. Let us assume, by induction, that they
are true for free n-step solvable groups. If F is a free group. then F(") is
a normal subgroup of F whose quotient goup F/F(") satisfies conditions 2
and 3 by the induction hypothesis. By Lemmas 2.5 and 3.9 we conclude that
the free (n + 1)-step solvable group F/:V("+l) sstisfies these conditions. O
Lemma 2.10 If F I N satisfies conditions 1 and 2, then cornmutution in
FIN' is transitive: f o r any nontriviul elements r , y, z E F/.V [x. y ] = 1
and [x, r] = 1 imphj [g, r] = 1.
Proof Let x. g, r E F/X' such that [x , y] = 1, [x. Z ] = 1, r. y. 2 # 1. I f
x E N / W , then by Theorem '2.5 y, 2 E !V/N'. which is abelian. Therefore
[y, z] = 1. I f x $ FIN' , then by the same theorem there are u, u E F/.V such
that x = IL", y = un, I = vk, z = u1 for some positive integers m. n, k , 1 . Since
F I N is an R-group by assumption, so is F/!V1. By 132, p. 2441 i t follows that
ttvo elements
both u and u
of F I N 1 commute if and only if their powers commute. But
are roots of r, so [u, .cl = 1. and hence [y. z ] = 1. as reqiiired.
In the rest of this section we will alivq-s assume that F / X satisfies con-
ditions 1-3.
Lemma 2.11 If x E F/IV1, but x @ N/W, then the centralizer of z in F1.V
u cyclic.
Proof Take an arbitrary element x E ( F I N t ) \ (N/VI.V1). By Lemma 2.9'
F/X ' satisfies condition 3, therefore x c m be represented in the form z = yk,
where y is a primitive element of FIN'. Let us prove that the centralizer
of x in FIN' is the cyclic subgroup generated by y. Indeed, suppose that
[x, z] = 1, then due to the transitivity of commutation in F / X f [g' r] = 1.
By Theorem 2.5, there exists an u such that y = um, z = u n for sonie m? n.
But y is not a proper power, so y = L*', and 2 = y" E (y).
Corollary 2.4 The centrulizer of an3 nontrivial element of F / N t N: u m u ~ i -
mal ube Ean subgroup of F / N 1 . Conuersely, any maximal abelian subgroup of
FIN' is a centralizer of some nontrivial element.
Corollary 2.5 If M is a maximal abelian subgroup of F/.Vt,, tlien either
111 = X/îVt, or il1 is ccyclic.
Lemma 2.12 In F / Y evenj maximal abelian subqroup of the f o n J I = ( g ) .
IJ $ N/iVr , is eonjugat e sepamt ed.
Proof Suppose for contradiction that for some x .II d l f n JI + 1, i.e.
there are mtn E Z such that x-'.ym+ = yn. Denote xdLyx by 2. Then
zm = y", therefore [zm, z/] = 1. Since [z7 zm] = 1 and commutation in F / X 1
is transitive, it follows that [z, y] = 1. This implies that z E .II. so ,- = g p
for some p. Without loss of generality, Ive may assume that p > 0.
The generators of the subgroup H = (x, y) satisfy the relation X - ~ I J X =
yP, so H is a hornomorphic image of the Baumslag-Solitar group Cl, = (a, b 1 a-l bu = b P ) under the epimorphism 9 : a ct z, b H y. Let us prove that q
is injective. -4s shown in [50], C i , z n.cl.(b) ~ ( a ) , where n.cl.(b) zz Z[b]?
(a) 2 2. Suppose that bias E kerip for bl E n.cl.(b); then ylxs = 1 for
some yl E n.cl.(y). But in the sequence yl, y,xs,x every two consecutive
ternis cornmute. By assumption x # d l , so [z, y] # 1, and by transitivity
of commutation Ive obtain that both y, and xS are trivial. This proves that
(x, y) = GL+ But z[;] E n.cl.(y) is an abelian subgroup of (2, y), so either
n.c1. (9) < N/N' (which is impossible since IJ # N / X f ) , or Z[$ is cyclic. which
is also false - a contradiction. [7
Now we are ready to construct the tensor completion of F / Y . Observe
thst FIN' is not a CSA-group, since the mauimal abelian subgroup K/!V' is
normal and hence not conjugate separated. But we can embed FIN' into a
ICA-group in the following way. Consider the group
Theorem 2.6 G belongs to the class Kr\.
Proof Let 61 be a maximal abelian subgroup in G. then by Lernma 2.1 .\,I
is of one of the following three types:
In case 2) X I is an A-module, and in case 3) d l is conjugate separated
by Lemma 2.3. Therefore it remains to consider case 1). Conjugating by
x-l if necessary, we may assume that M < FIN'. I f d l = NIN', M is an
A-module, so we cm restrict ourselves t o the case M = ( g ) , y E F/!Vf,
y NIN'.
Suppose that xdLymx = yn for some z E G. PVrite x in reduced fom:
x = nl fi nJ,, where ni E (NIN' $ A), f* E F/N'. Then we have the
equality: f ;ln;' ymn lh n,f, = y". Cancellation in the Ieft-
hand side can occur only if n;lymnl in the amalgamated subgroup !V/!V1,
which by the normality of N/N' implies that y" E N/!Vt. Then in FIN
y" = 1, and 3 # 1, but FIN is torsion-free by assumption. Therefore nl = 1.
Proceeding by induction, we obtain that al1 ni = 1, therefore x E F / W . But
in FIN' the subgroup (y) is conjugate separated by Lemma 2.12. The proof
is complete.
Observe also that in any of the three cases hl is faithful over -4. Cl
Summing up? we have embeddings F/.V1 v G v Gd4, and since G E
Kt\ and al1 maximal abelian subgroups of G are Faithful over -4. the tensor
cornpletion of G is the complete tensor extension of centralizers of G by the
ring .A.
Theorem 2.7 (F/LV')*~ = GT'
Proof Let i l : F / Y v G, i2 : G v Ge4. Take an arbitrary A-group H and
a partial -4-homomorphism 9 : F/M1 -t H . By the universal property of
a free product with amalgamation, there esists a partial A-homomorphism
pl : G + H such that rp, o i l = p. Then by the definition of G..' there exists
an A-homomorphism $J : GA -+ H such that <pl = ti, O il, and p = $J 0 i2 O il.
Therefore GA satisfies the universal property of the tensor completion of
FIN'.
It remains to show that FINt A-generates GA But it is clear that FIN'
generates G as a partial A-group, and G A-generates G A by definition.
Hence GA = ( F / N t ) ~ . u
Corollary 2.6 A free solvable group iCI. = FIN' , where N = F("-')? is
faithful over a n y torsion-free ring -4, and the tensor completion of !\In can
be constructed as the complete tensor extension of centralzzers of a certain
partial .L -group containiny Al,, .
Corollary 2.7 T h e tensor completion of a free soluable group need not be
soluable.
Proof Let iV = F("-'). The tensor completion of FIN' contains the
subgroup G = FIN' * ~ p p ( . i /N1)" , which arises on the first step of our
construction. If FIN' is nonabelian, then G, in its turn. contains a noncyclic
free subgroup and thus is not solvable. a
3 Equations in the Q-complet ion of a torsion-
free hyperbolic group
3.1 Introduction and basic definitions
The problem of algorithmic solvability of systems of equations is closely con-
nected with the farnous Tarski problem about the decidability of the ele-
mentary theory of a free group, as well as with the developrnent of algebraic
geometry over groups. Therefore, the study of systems of equations is one
of the main streams of modern combinatorial group theory. One of the most
significant results in this area was proved by &Iakanin (-131 and Razborov
[56], who established the algorithmic solvability of systems of equations over
free groups. Rips and Sela [61] solved systems of equations over torsion-free
hyperbolic groups by reducing the problem to free groups. Kharlarnpovich
and hfyasnikov [25] solved finite systems of equations over the free Q-group
FQ (the tensor Q-completion of a free group F). In this chapter Ive prove
that an even more general fact is true: finite systems of equations over the
Q-completion GQ of an arbitrary torsion-free hyperbolic group G with gen-
erators d l , . . . , dN are algorit hrnically solvable. The results of this chapter
are accepted for publication (7-31.
Main theorem Let G be a torsion-free hyperbolic group. Then there exists
an algorithm thot decides if a given finite systern of equations over the Q-
cornpletion of G hus u solution, and if it does, finds a solution.
By a triangular equation we mean an equation with a t most three letters.
Given a system S of equations in GQ, we may assume that al1 equations
in S are triangular. Indeed, an arbitrary equation x;'x;' . . . xEn = 1 (where
2, stands either for a variable or for a constant, ~i = f 1) is equivalent to
a finite systern of triangular equations: = 1, y&.y.' = 1. . . . . En-1 E R =
~ - 2 2 , - , x, 1. Adding a finite nurnber of nem variables and equations. we
may also assume that S contains only equations with coefficients in G. To
achieve this, we replace every constant of the form d z by a new variable ,-
satisfying the equation zn = dm.
From now on, we fis a finite system S of triangular equations over GQ
with coefficients in G. We will reduce the system S to a finite set of systems in
a specific hyperbolic group G * ( t ,, . . . , t,(,)), where the number ~ ( m ) can be
determined effectively given the number n of equations in the original system
S. The resulting systems are accompanied by the restriction that some of the
variables belong to a certain subgroup G * ( t i t . . . , t,) of G * (t l . . . . . t,(,)) Le.
do not contain certain t t s . To each of the systems we can apply the (slightly
modified) method of Rips and Sela (611 to see if they are decidable. If none
of them is consistent! then the original system S has no solution: if at least
one has a solution, then it is possible to find a corresponding solution to the
system S.
Below we describe the Q-completion GQ of a torsion-free hyperbolic group
G as the union of an effective chain of hyperbolic subgroups (details can be
found in (241).
For an arbitrary torsion-free hyperbolic g o u p R and natural number
n 2 2, choose a set of elements V,, = {ul, . . . , ut,) E R satisfying the following
condition (Sn) :
1) V, consists of cyclically minimal (of minimal length in its conjugacy
class) primitive elements of length not more than n;
2) no two centralizers in the set of centralizen {C' ( tu) , v E V,) are conju-
gatc in R;
3) the set Vn is maximal with properties 1) and 2) , i.e. any elernent of
length not more then n is conjugate to a power of some ç E V,.
By definition,
Notice that this definition does not depend on the order of elements in Vn.
It was proved in [NI that GQ is the union of a c h i n of hyperbolic groups
mith T, = Tn-i (V,), where V, satisfies the condition Sn in the group Tn-l.
Definition 3.1 Let a group H be an amalgamated piodauct H = S +,=tr (t) .
Then t-syllables of the vord botatbita2 . . . b,, zuhere 6, E S? are the sub-words
ta', . . . , t a n . If u is an element in H, then Irul is the number of occurrences
of t-syllables in a reduced word representing u. .ive call this number the t-
length of u.
Suppose that the system S has a solution in GQ; let {'CL, . . . , ,YL) be
a solution with the minimal possible number of roots. Since the solution
belongs to GQ, it is contained in a certain goup K, obtained from G by
adding finitely many roots. It is the union of a chain of subgroups Hi defined
as follows. Let G = Ho
Step 1. Consider pairwise nonconjugate cyclically minimal primitive ele-
e t u , . . . , , G u . . . - < i ~ l ~ , l (here lu1 denotes the l~ng t l i u l cr
in G) , and add roots t l , . . . , tt,, S U C ~ that Uj = t:' . (Notice chat 'uiCl does
not becorne a proper power after we add roots t l , . . . , t,.) The corresponding
groups are denoted by HI,. . . , H k L , where Hj+i = H,+
Step 2. Consider pairwise nonconjugate primitive elements u k , , ~ ? . . . , L L ~ ? E
HI, cyclically reduced in the amalgamated product , each having the reduced
form u = t?'cl . . . t;lkcI, where ai # O. ai E 2, , ci E G, l ~ ~ ~ + ~ l H 1 5 . . . 5
lukz 1 H~ ; and add roots tnl+l, . . . . t k , . such that 'uj = t:' to the group Hc, . The corresponding groups are denoted by f & l + I ? . . . . Hk2, where H,-l =
Step i + l . Suppose that H I , . . . , Hk, have been constructed.
Consider painvise nonconjugate primitive elernents u k l ? . . . , Uk,,, E Hil
cyclically reduced in the amalgamated product, each having the reduced
form tP1c1.. . t l k c k , where ai # O, ai E Z, , E Hidi ( cr. is not a power of
ui, because the elements are cyclically reduced), luki+ 1 < . . . 5 Iuk,+, 1 H, ,
and add roots ,..., tk,+,, such that uj = tiJ to the group Hk,. The
corresponding groups are denoted by H k i f l , . . . , Hki+,. where
Finally, for some nurnber i one has = Hki+i.
The group Hki+, is called the group at level i l corresponding to the se-
quence u l , . . . , uk, , , . The group H, will be called the group of rank i. GVe also
order the set of tjls: ti < ti if k < 1.
Definition 3.2 An element in Hi+, is said t o be d t t e n in reduced form
in rank i if it belongs to Hi and is in reduced fonn as an element in the
ama~gamated P T O ~ U C ~ Hi = Hi-l *,i=,;, (ti). If r = O. then the element is
said to be in reduced / o n in al1 ranks 2 i if it is in reduced fonn in runk i.
An element h in Hl+, is defined by induction on r to be suritteri in reduced
form in al1 ranks 2 i if it is written in reduced form in the arnalgumated
product HiCr = * - t s i + r ( t i+ t ) i h = bot~:,blt~., . . . b,, where the ut+?- ,+r
I l , . . . ,b, E Hi+,-l are in reduced fonn in al1 r a d s 2 z.
bIany of the proofs in the subsequent sections will involve the method of
Van Karnpen diagrams. Below, follotving [25], me give some basic definitions
related to this method.
Recall that a map is a finite, planar, connected 2-coniplex. By a diugmm
A mer a presentation (ni , . . . , a,l Ri . . . , &), where the words R, are cycli-
cally reduced, we mean a map with a function $J which assigns to each edge of
the map one of the letters a:', 1 5 k 5 rn, such that $(e-l) = (qj(e))-l and
if p = e l . . . ed is the contour of some cell 9 of Al then $ ( p ) = 4(ei) . . . d(eJ
in the free g o u p F ( a l , . . . , a,) is a cyclic shift of one of the defining words
The word d ( p ) is called the label of the path p. The label of a diagram
A (whose contour is always taken with a counterclockwise orientation) is
defined analogously.
Van Kampen's Lemma states that a word CC' represents the identity of
the g o u p G if and only if there is a simply connected (or Van Karnpen)
diagram A over G such that the boundary label of A is CL*.
Let A be a diagram over H,. A t,-strip is a subdiagam with the bound-
ary iabei t;:"uyE (see Figure O(&)), consisting of cells ivith the boundaq-
t S J , ü L , J j which are termed tj-cells (see Figure O(b).) Two +trips can be
glued together to form a paired t j -s t f ip (see Figure O(c).)
.s,.
Y~
Figure O
Let Fi be the free group r i t h the same generating set as H,? i = 1,. . . . k i
and let p : Hi -+ F, be a section, i.e. a mapping of sets such that r O P =id.
For Our purposes, P has to be of specific form; we will define it explicitly in
section 3.3. For ,Y E Hi Ive will cal1 B(X) the canonical representative of S.
Definition 3.3 By an equational diagram over a g r o ~ ~ p Hi we mean an equa-
tzon X i . . .,Y, = 1 together 102th c solution A i , . . . , -4, and a diagram over
Hi havzng P( i l i ) . . . @(A,) as i t s boundarj label. A n equational triangle zs an
equational diagram corresponding t o a triangular equation. A system of equa-
tional diagrams zs a sys tem of equations together wzth the sys tem O/ diagrams
svch that the solution associated t o each equational diagram is a solution O/
the whole systern.
A free equational diagram i s a n equational diagram v z t h no cells: the
corresponding equations are cclled free equations.
If the level i of the group Hki+, is greater than zero, we can use the method
of (251 to reduce each equation of the system S to a bounded nurnber of free
equations and at moût one equation over the group in the previous rank. We
can proceed in the same way until WC reach level O (i.e. the group I f k , ) . for
which we need to develop a separate method. In whnt follows Ive will denote
Hk, by H .
3.2 Some properties of the Cayley graph of H
Let us recall briefly some basic notions from the theory of hyperbolic groups;
for more details, see, for example, [Ml.
Definition 3.4 Let lx- y1 denote the distance between two points in a metric
space X . The Grornov product of points x, y E S with respect t o a third point
o E X is by definition
1 (x * y). = ?(lx - 01 + Iy - 01 - lx - YI) .
T h e space X is called 6-hyperbolzc with respect to O, if there exists 6 2 O svch
that for dl x, y , x E .Y
(x y). 2 min((. z) ,: (y r).) - 6.
Note that the property of being hyperbolic does not depend on the choice
of the reference point O:
Lemma 3.1 [I?] If S is 6-hyperbolic with respect tu o E X. then it is 2<)' - hyperbolic with respect to eue? point x E .Y.
-4 metric space S is called 6-hyperbolic if it is d-hyperbolic mith respect to
every point x E -Y. S is called hyperbolic if such b 2 O exists. An important
example of a hyperbolic space is the Cayley graph C(G) of a hyperbolic group
Lemma 3.2 [54] If G is a 6-hyperbolic group, then its Cayley graph C(G),
viith the rnetric defined bg lg - hl = 1i~-lhl, is a cTr-hyperbolic spuce for 5' =
d + 2 .
CVe will frequently use the following lemma:
Lemma 3.3 ([54: Lemma 1.51) For each geodesic triangle [ x l , 22: x3] in a
6-space, there are points E [xi- , (indices are considered modulo 3)
such that
d(xir pi-1) = d(xit y i + ~ ) = (xi-1 * x i + i ) ~ , i
d(yi , yi- 1) 5 46 and d (SU' [ri, yikl]) 5 46
for any point u E [x i , yikl]. We will cal2 the triangle [yl, 92, y3] the Gromov
triangle inscribed in [x 1, 2 2 x3] (see Fig. 1).
Definition 3.5 In a rnetn'c space .Y, consider u path p with natural pura-
metriration by length. The path p is called (A ! c)-quusigeodesic. if there erist
X > O and c 2 O such that for antj points p ( s ) and p ( t )
Lemma 3.4 [54, Lemma 1.91 There exivts a constunt ,u = pi& A. c ) such
that for any (A: c)-quasigeodesic path p in u 3-space and uny geodesic path q
with the same initial and terminal points as p! the inequalities d ( Q . p ) < ~i
and d(P, q ) < ,u hold for uny points Q E q und P E p .
Definition 3.6 Let u, v E {u l : . . . , u k , } , 1 u l G l l v Ic. In r ( H ) consider two
paths: a u -puth 7-1 and a -path r*, where r l connects the sequence oj vertices
. . . , g ü l , g, gu, gu2,. . ., r2 connects the sequence of vertices . . . , ghv-l. gh,
ghu, ghu2,. . ., h E H, g E G. (See Fzg.2.) Then a path q that connects rl
and r2 is called a minimal path if
1. the image in H of the label of q is an element in reduced form in al1
mnb from O to kL;
2. for each subpath q' of q connecting t-syllables, the image of the label
of q' is a geodesic word in G and has minimal length amony al1 wordv
connecting the corresponding t -strips.
Figure 2
Definition 3.7 Let d(P, q ) denote the distance bet*ween u point P and a path
q in the Cayley graph of G. Then a point P' is culled the projection of P
onto q iJ P' E q and d(P, P') = d(P, cl).
The following lemma says that the initial (resp. terminal) points of an?
two minimal poths are, in a sense, close to each other.
Lemma 3.5 Let u, v , r l , 1-2 be as in Definition 3.6, with rl and distinct;
denote by U the cyclic subgroup of G generated by u, and by I r the cyclic
subgroup generated by W. Let q and q' be two minimal paths connecting ri
and Q, X the u-periodzc word corresponding to the subpath of rl connecting
the initial points of q and qf, and k' the u-periodic word corresponding to the
subpath of r2 connecting the terminal points of q and q'; ie.,
Then there is a constant hl = 111 (G, U, Lw! 1 u 1,1 u 1 ) such that
Proof Whenever the context allows us to do so. we will often use the sanie
notation for a path in r ( H ) and the corresponding word in H.
We may assume that q and q' do not contain any t,'s (otherwise subdivide
the diagram corresponding to the equality q-LSq' into sereral subdiagrams
A l , . . . , A, with side labels containing no t-syllables as s h o w in Fig.3: for
each of them obtain a constant :Ut; take A I to be ma?<(&. . . , .\ln) ) .
Figure 3
Observe that the subgroups Li and \ - are quasiisometrically embedded in
G; therefore, the paths corresponding to S and Y are ( A 7 c)-quasigeodesic
for some X > O and c 2 O (see [SI). Let SI and Yi be the geodesic paths
having the same initial and terminal points as the quasigeodesic paths .Y and
Y .
In the geodesic quadrangle ABCD, draw the diagonal BD. Inside each of
the resulting triangles consider the Gromov triangles from Lemma 3.3. The
triangles can be located with respect to each other in two different ways. as
s h o w in Fig. 4 a) and b).
Figure 4
Case a). Let Di be the projection on CI ont0 B2 D and D2 the projection
of C2 ont0 BIB. Then by Lemma 3.3 d(B lo Ci) 5 dB. 4 C i . DI) < 4d.
d(&, C2) 5 4 4 d(C2, D?) < 46: t herefore
Now take the projection of BI onto S and let El be the phase vertex
on S closest to this projection; in the same way, obtain the phase vertices
E2 E .Y and FI, F2 E Y. Then d(Bl, E l ) , d(&, f i ) , d(D2, E2), d(& F2) are
al1 less t han p + mau(l u I l l u 1). If p is the pat h connecting El wit h FI and p'
the path connecting E2 with F2, then 1 p I l l p' 15 86+2(p+max(( u I,I u 1)) .
If u # V , the subgroups U and V satisfy the conditions of Lemma 4 from
[24]: they are quasiisometrically embedded in G, malnormal. and
U n g-iVg = ( e ) for al1 g E G by the choice of u and u. Therefore. there is
a constant i\fo = Mu(G, U. 1') such that either mas ( ) ELE2 1 . 1 FLF2 1 ) < -\Io
or .Lmau(l p 1 ,I p' 1) 21 EIEz 1 . 1 F1F2 1. Since in our case p and p' have
bounded length, it follows that
If u = *u, 1 ELE? )> Mo, then by the same Lemma either / El E2 1. 1 FL F2 15 4 mw(l p ) , I p' 1) or p, pl represent powers of u . In the latter case.
the paths r i and 7-2 coincide. which is the case that ne are not interested in.
It remains to bound AEl, BE2,CF2 and DFL.
Observe that 1 El BLdL 15 .L<i + p+ ( u 1; if .4.-LL had greater length. the11
.4Al D would not be a minimal path: ElBl.4iD , also connecting rl with r - ,
would be shorter. Therefore, 1 -4.4 1s -45 + p+ ( u 1 . and 1 .-LEL 151 .-ML 1 + ( El BIAL 15 86 + 3p + 2 1 u 1, as required. The same argument applies to
the remaining paths.
Case b). Let DI be the projection of Ci ont0 B.42, LI2 the projection of
C2 ont0 DA1. Observe that ( EIBICIDi (=I ElBL ( + 1 BlCL + i CID1 (5
p+ 1 u 1 +86. Therefore, 1 BDI 15 p+ 1 u 1 +8S (othenvise BC = q' is not a
minimal path), and 1 E I B 151 EIBICIDi 1 + 1 BRi 15 2(p+ 1 u 1 +S6). In
a similar way, 1 AEl 15 2(p+ ( u 1 +46), which implies
The same argument works for Y.
It follows from the above that the constant M can be taken equal to
rnai<(i~Io, 4(86 + 2(p + m m 1 IL 1,I v 1) ) ) + 2-18 + 4p + 4 1 zl 1. 0
Therefore, for u, u E {ul,. . . , uk,), a u-path r.1 and a wpath r . ~ from
Definition 3.6 there are subpaths r', c rl and r i c r2 such that 1 ri 1 . 1 r; 15
M, the initial points of al1 minimal paths connecting ri and rp belong to r i ,
and the terminal points? to r i .
Definition 3.8 Folloving the tenninologg of [25], .we cul1 u minimal path
between ri and r2 a (uov)-pseudoconnector for h. Let q be a (u.u)-pseudo-
connector, pi = ~ ( q ) , p- = ~ ( q ) ( pl E r i , p? E ri) . Consider the two phase
uertices bl and b2 closest t o pi o n each side of pz on the u - p d h r?. The
connecting zone for h with respect to g zs the union o j al1 phase vertices
between such bl and b2 f o r al1 (u,v)-pseudoconnecto'rs for h. Denote the
initial uertex O/ the connecting zone bg ( h ) 1 and the terminal ver ter by (h)? .
3.3 Construction of canonical representatives
Let F be the free group with the same set of generators d l , . . . . d n as G: let
Fi = F * ( t l , . . . , t i ) . For O 5 i 5 k k me will define a section 3 : Hi + Fi (a mapping of sets such that n O ,û = id). For -Y E Hi we mil1 cal1 @(-Y)
the canonical representstive of X. (See [25] for the construction of canonical
representatives for elements in the groups of ranks higher than kl. )
The canonical representative of an element X E G is simply a fixed
geodesic word representing S. If X is an element of Hi for 1 5 i 5 kl and
S contains t,, then it can be written in the form .Y = bot: blth - tzb,,
where ij E {l,. . . , i}? bj E G. By E j Ive denote the canonical representative
of the path connecting the vertices ( b y L ) * and (b j )?; Ive will cal1 6, the (IL' u ) -
pseudoconnector for the elernent bj (here -u = uj-1, u = u j ) . The points
( b j - L ) I and ( b ~ ' ) ~ c m be connected by a path with label t:. Then the -
canonical representative of -Y is defined to be 6,t::6,tb . . . b,L.
In the example illustrated in Fig.5, S = botLLbls-ib2, where u .c E
{ . u l , . . . , Z L ~ , } , u = t 3 , u = s2. Then O ( S ) = .(&)t-5,8(b;)s-3,3(&).
Figure 5
Below, al1 the elernents ,ujTs are always representeci by the words ,J(u,)
and we will write ~ l j instead of @(ruj). It will be clear from the contest when
uj means a word and nhen it means the element represented by this word.
3.4 Middles
Let X = { S i , . . . , Kt} be a solution of the systern S in the group GQ.
Suppose that this solution is minimal, Le. contains the minimal possible
number of roots. Then for some i the solution -Y belongs to the group
For the solution X we will construct another solution SI ,' . . . . Si and a
system of equations over the group G * (tl,. . . , ti,+,) such that ,d(S;), . . .
f i(Si) will be a part of a solution of this new system, and every solution of
the new system will give a solution of the system S.
Here we will deal only with equational triangles in the group H = Hk,.
since for the higher ranks we can use the reduction method developed in [25].
Consider an equational triangle in H with at least one side label contain-
ing t i for some i E {l, . . . , kL) (hence at least two side labels containing ti).
It is represented by a diagram of the form shown in Fig.6. 'lote that the
t-strips in Fig. 6 rnay correspond to distinct t's; a &+trip c m start on a side
of the equational triangle, but not on a uj-side of a t,-strip. since u j E G
and hence doesn't contain any t's.
The following lemma is a direct consequence of the construction of canon-
ical representatives; it iç an analog of Lemma 3 from [%].
Lemrna 3.6 Suppose that in H we have o diagram (see Fig.7) wdh the P. rjn+t
bounda y label (blt j i l . . . tz bntin+,)tj,,+, (cltZ1 . . . ~ ~ t ~ ~ + ~ ) - ~ , where j l , . . . ,
jn+i E 11,. . . , Ici) , bi, cl, . . . , b,, % E H. ~ h e n ,8(b1th1 . . . t Y n l i t b t 3n+i )t""" 3n+i =
Figure 7
Figure 8
It folloms from Lemma 3.6 that every equational triangle either does not
contain any cells or takes on the form shown in Fig. 8 and hence has a unique
mavimal nontrivial G-subdiagram. (Some of the t-strips in Fig. 8 might. of
course, be trivial.)
Definition 3.9 Consider an equational triangle 11 in i'(H,) (1 < i < kl) with the bounda y label @(X1)/3(rb)P(-X3). The maximal nontrivial G-sub-
diagram of 4 is called the middle of this triangle. For example, the subdia-
gram A B EFDC in Fig.8 is a middle.
Let u E {ul ,... iuk.), t E { t l t k i } . The boundary of a middle is
canonically subdiuided in to paths. Each of these paths i s ei ther a u-path join-
ing two phase vertices (i.e. lelongs t o the u-side of a t -s t r ip) or a connecter.
The u-paths are called pseudoangles of the middle.
Let v = ma?<{vl?. . . , y), where Y , are certain jixed constunts dependzng
o n the group G and ul: . . . . up,; we will see how tu calculute the v, 's in the
proo f of Lemma 3.7.
A pseudoangle p 2s said to be trivial if the corresponding u-puth is trivial;
short zf 1 p 1 5 u ; long o t h e m i s e . A rniddle is called triangular if al1 the
pseudoangles are trivial.
If none of X, 's contains t then the rniddle of the triangle coincides with
the triangle itselj and is a triangular middle.
As an example. consider the middle AB EFDC in Fig. S. The paths
.AB' CD, EF are pseudoangles of this middle.
Each middle represents an equation over the group G; denote the corre-
sponding systern of equational diagrams by 3.
Lemma 3.7 Let AI be a middle of an arbitranj equational triangle over H .
T h e n at rnost one pseudoangle of M is long.
Proof We will provide a proof for the case where none of the pseudoangles
is trivial. Take an arbitrary triangulation of the geodesic hexagon obtained
from the middle M by replacing the pseudoangles, which are quasigeodesic
paths, by the corresponding geodesic paths. CVe will use the triangulation
shown in Fig. 9. Here A B is a u-path, CD is a v-path and EF is a w-path
for u, v, w E {q , . . . , ut, ). Inside each of the resulting triangles ABE. ACE.
C E D and DEF consider the Gromov triangles h m Lemma 3.3. These
Gromov triangles can be arranged with respect to each other in eight different
mays. Instead of providing a detailed proof for each of the eight possibilities.
we will consider two typical cases; the others c m be analyzed similarly. Sote
also that if one or two of the pseudoangles are trivial? then instead of the
geodesic hexagon we obtain a pentagon or a quadrangle, which leaves us with
fewer cases to consider.
Figure 9
For an arbitrary path q in the Cayley graph of H. ne denote by <7 the
geodesic path with the same initial and terminal points as q. If Pl is the
projection of a point P ont0 a path ql and P2 is the projection of Pl onto a
path 92, then we will cal1 f i the projection of P oiito q? through 91.
Case 1. (Fig. 10 a).) Let .A4 denote the projection of Ai E .= onto AB,
C4 the projection of CI E onto CD and D4 the projection of DI E TF onto EF.
Claim: &B is short.
Proof. We may assume that 1 A I B I> JI (the length of the connecting
zone for BE); otherwise there is nothing to prove. Let p be the minimal
path between AB and EF whose initial point is the larthest from B; r ( p ) is
between A., and B. The path BE is a connector, so its length is bounded by
the following: 1 BE 15 1 p 1 +2M.
We nonow claim that 1 B.& 151 .44.4L-42 1 +.)M. Indeed. suppose for
contradiction that ( BA2 1 > 1 .L4rli.-ls 1 i 2 M . Consider the path BA2 E.
path between .AB and EF, but its initial point falls outside the connecting
zone for BE - a contradiction.
Therefore 1 BA2 151 .4+-il 1 + 1 .&.& 1 + 2 M 5 p + 46 + 2.11; by
Lemma 3.3 1 .A1B I=I BA2 (5 ,U + 4S + 2M: finally, 1 .-14B 151 .-l.l.41 1 + 1 .A1 .A2 1 < 2p + 46 + 2 M , as required.
Claim: .4A.4 is short.
Proof. Let As denote the projection of .-13 ont0 .AB1; by the same argu-
ment as above ( 15 -44.41.&.& 1 +?XI (otherwise AC is not a connec-
tor) . Consequently, we have: 1 .A.& 1 < p + 85 + 231, and 1 .4.-LI 15 ( .4.-15 1 + 1 .&.41.43.& 15 2~ + 16s f 2.bf.
Combining the two claims. we obtain that .AB is a short pseudoangle.
In the sarne way one can see that CC5, C I D , FD.1 and D.&' are al1 short
(here Cs is B3 projected onto CD through then onto CCl). However, i t is
not guaranteed that C5C4 is short? so it is possible for the pseudoangle CD
to be long. Fig. 10 b) gives a more precise idea of what the middle looks like
in this case.
As one can see from the calculations? ( AB 1, 1 EF 1 5 4(p + 106 + AI):
denote this constant by vl.
Case 2. (Fig. 11 a) .)
Let Dg denote the phase vertex closest to the projection of D2 onto CD
through DC1; C4 the phase vertex closest to the projection of Ci onto CD;
4 the projection of .A2 onto EF through B& B3E, C3E and EDi.
In this case, AB is short by the same argument as above; it is also clear
that the following are short: CC4, DJ, ED6 and DD5.
Claim: C4D5 is short.
Proof. Let Di E EF be the phase vertex closest to the projection of C3
ont0 EF through EDI; Ds the phase vertex in EF closest to the projection
of Dl. Denote the path C4CiC3D7 by p and D5DaDiDs by q. Then by [25,
Lemma 41 ( D7, DB 1, 1 C4 D5 1 < max(Mo, 1 p 1) 1 q 1) for some constant &.
(Note that neither v nor participates in p or p.) Since 1 p / and 1 q 1 are
bounded by 2 m w ( J u I,I m 1) + 2p + 85, this gives us a boundary on the
lengt h of C4D5.
D6D7 might turn out to be long. See Fig. 11 b) for the shape of the
middle in this case.
These cslculütions give us L he cous tari t v* huridirig the lerig tlis üf the
short pseudoangles AB and CD:
In a similar w q , one can compute the constants u3,. . . , u8 bounding the
lengths of the short pseudoangles in the remaining cases. 0
The following lemma can be proved by the same method as Lemma 3.7.
Lemma 3.8 There are constants i\Ii = Mi (G, I L I , . . . . uki ) and
-4 = M2(G, ui, . . . , uk l ) such that for euerg middle ABEFDC (Fig. 12)
u~ith a long pseudoangle CD there ezist points Pl Q E B E and phase uertices
Pl , QI E DC with the following properties:
d(Q1 Q I ) < MI und for all P' E PQ d(P' , PIQI) < Ml:
Proof Let us take the same triangulation of the middle AB EFDC as in the
proof of Lemma 3.7. We will provide a proof for the case where the Gromov
triangles inscribed in the triangles ABE? AEC, CED and DEF are located
with respect to each other as s h o m in Fig.10 a); denote by hl,! and the
constants satisfying the conditions of the lemma in this case. Similarly. one
Figure 12
can consider the remaining seven cases to obtain the constants AI:. i = 1.2.
j = 2 , . . . ,5. Then put dll = rna?c(M:, . . . , .\le), = rnau(.\&. . . . .Le). In the notation of Lemma 3.7, let P be the projection of Cz ont0 EA2
through EB2. Q the projection of BI onto P.&: and Pl. QI the phase vertices
on CD closest to C4 and Cg, respectively. Then d(P. Pi). d(Q. QI) 5 12s + p+ma?t(lull ... ., IukiI) = M:.
In the same way as in the proof of Lemma 3.7, one can see that ICC5(.
IC4D1 5 165 + 2p + 2A1. Dividing this constant by rnas( l -ul 1 . . . . . luk, I ) .
taking the integer part of the resulting number and adding 1, we obtain the
desired constant i'C1;. O
3.5 Shrinking of the middle t-strips
Our next objective is to bound the powers of 'Uj for j = 1, . . . , kL participating
in the long pseudoangles. This will allow us to bound the number of trian-
gular equations over the group G obtained from the middles by triangula-
tion. (For instance, the middle with the bounday label @(s i)/3(u$?(sz).0(s3)
shown in Fig. 13 produces 1 + 1 triangular equations.)
Figure 13
'laking the middles of al1 equational triangles over the group H corre-
sponding to the system S, we obtain a system of equational diagrarns; let us
denote this system by 3. Eacli diagram of the system 3 has the boundary
label of the form ,û(Sl) . . . $(.Yi). In what folloms, it ni11 be convenient to
ternporarily replace ,8(Si) for sorne of the Si by the canonical representative
of .Yi in the sense of E. Rips and 2. Sela ([6l]). (W will explain precisely for
what -Yi we do this replacement after we define shrinkable words later in this
section.) Let us recall the definition of Rips-Sela representatives. follosing
the interpretation giveii by T. Delzant in [Il].
Let D be an integer, : (ai. . . . .a,; Ri.. . . , RT) a trinngular systern of
equations in a torsion-free hyperbolic group G. and al. . . . , a, a solution.
- 1 , Denote a;' = a-il cri - O-,; the equations are written in the form rt,,,k :
a,ajak = 1, mhere i l j, k E {+l, . . . , z b ) .
Definition 3.10 A farniiy of D-canonical representatives in the sense O/ E.
Rips und 2. Sela of ai, . . . , a, is a family of quasigeodesic pol~/gonal paths
..ii(t) of length Li in the Cayley gmph r(G) of G (.Ai(t) : [O, L,] + r(G),
Li E N , ai = &(Li ) ) porametrized by arc length and satisfyzng the condition
bel0 W.
Set A-i(t) = a-iAi(Li - t ) ; let r i j k : <jajaç be one of the equations. Then
Figure 14
the triangle .A,, ai.Aj. CkiCkj.Ak is D -Jat (see Figure 14) in the fdowing sensr:
The paths .Ai are quasigeodesic, in other words,
where v, 5 (2iV)' denotes the number of elements in a hall of radius r , iV is
the number of generators of C.
Let A' denote the constant l/.uloa. It follows from Lemma 3.4 that there
exists a constant pl = ~'(6, A') such that for every geodesic word o and its
Rips-Sela representative A the inequalities d(Q, -4) < p' and d(P, a) < pl
hold for any points Q E a and P E A.
The main technical result of E. Rips and 2. Sela ([61]) consists in the
following.
Theorem 3.1 [61] Let T be a positive integer. Then there exzsts a number
D depending on19 on T and the generutors O/ G such that for euerg s~ystenr
.!? of T triangular epuations over G , each solution al . . . , a, of !? admits a
familg of D -canonzcul representatives.
Sow let us return to the systern of equational diagrams over G obtained
by reduction from the original system S.
Definition 3.11 A n d d l e t, +trip is a t, -strîp whose u, -side belongs to the
long pseudoangle of a rniddle of a n equatiortal triangle.
Definition 3.12 A subfword IL? of the lubel of the boundary of a middle is
said to be shrinkable if one of the foZlo,winy conditions is satisfied:
1. u zs the Inbel uf the u-side of u rniddle t -strip (e.g.. C D in Fig. 10 b):
2. w is u connecter such thut a subword t.' of w is "close" t o the u-side q of
a middle t -stript meaning that for every point P E v d ( P , q ) 5 .\il + p'
(e.g., BE in Fig. 10 6);
3. w as a t-periodic word on the t-side of a middZe t-strip;
4. if w is the label of a cornmon subpath of two paths p and p' shnnkable
as a sub#word of the label of p, then it 2s shrinkable as a subword of the
label of p'.
Figure 15
For each j = 1,. . . , kl and each middle tj-strip, we will replace the power
of ilj on its boundary by a bounded power of u j ; during this procedure, the
solution might be altered, but it will remain a solution of the original system
of equations.
When ive replace iz u,-nord u: by another word. rve should rnake a p p -
priate changes to the diagram whose boundary contains the word u. as well
as to the adjacent diagrams. Consider, for example, the diagram .-LBCED
s h o w in Fig.15 a): here CD is "close" to AB in the sense of Definition 3.17.
If we remove a subword w from .AB, Ive will also have to remove a subword
of CD and a subword of DE (or EC) together with some cells. Similarly. iri
the diagram ABCFED in Fig. lsb), removing a subword of AB forces us to
remove some subwords of CD and DE (or CF). In this diagram me assume
that both CD and EF are close to AB, and EF is a short pseudoarigle of
the diagram CDEF (if it nere long. it could not be close to AB by Lemnia
3.5).
For the sake of convenience, we will non denote by B ( S , ) the Rips-Sela
representatives for the words S, that are on the b o u n d a ~ of triangular di-
agrarns from the systern (e.g., CED frorn Fig. 15 a)) and the diagrams
obtained by triangulation from polygonal diagrams containing no long pseu-
doangles ( CDEF in Fig. 15 b)). The diagrams frorn Fig 15 a).bj now take
shape shown in Fig 15 c),d) respectively.
Definition 3.13 L e t 0 be the least i n t e g e r greuter than
31\f2 + 2 0 1 ma.x(lul 1, . . . , Iuki 1). The length L ( w ) of a shrinkable word u: zs
defined os follows.
3. Let u (or a subeword of tu) be close tu the uj-side O/ u n i d d l e t , -
strip in the sense u j Dr/initiori 3.12. Drriutt! l y F uiil Q ilic p luse
vertices closest to the projections of L ( u ) ) and ~ ( w ) onto q' respcctivelg;
the path PQ corresponds to a certain subzuord US of q. Then we put
L(w) = k - o.
Definition 3.14 A maxzmal shn'nkable subdiagrarn A O/ an etputional tr i-
angle zs the minimal subdiagmm haviny three mazimal shrinkable words on
the boundary.
In Fig. 16 we illustrate al1 passible shapes of mavimal shrinkable subdia-
grams. (The diagram ABC in Fig. 16 ci) is shrinkable since the path AB is
'tlose" to a u-side of a middle t-strip.)
Every maximal shrinkable subdiagram Ak gives us an equation over the
integers as described below. Let ski, ak? and a k 3 be the maximal shrink-
able wvords on the boundary of Ak and let u be the element from the set
{,ul?. . . , u k , ) used in the definition of length for the mords a k j ; denote L(ak,)
by l k j . (bve will be only interested in the case where at l e s t one of l k j is
positive, since we do not need to "shrink" any words of nonpositive length.)
Then to the subdiagram hk we associate the equation lkl + lk2 + o + s = lk3.
where s E {O, . . . , Dr) and D' is the least integer geater than D/lul. (For
euample, if Ak is the diagram shown in Fig. 16 a), then the tj-1vord AD is
of length l k l , the tj-word EC of length Z k 2 and the * U ~ - W O T ~ A C of length lk,;
we obtain the equation l k l + l k l + 0 + l = IL=.)
Figure 16
We obtain an equation of this form from every maximal shrînkable sub-
diagram; denote the system of these equations by L.
It is possible to obtain only a finite number of distinct linear systems of
this form. Every such system is algorithmically solvable, since the elementary
theory of the natural numbers with addition is decidable. Choose arbitrarily
a solution for each systern. Let 1 be the maximum of l i j in these solutions.
Then the systern L also has a solution bounded by 1. Let
be this solution,
We now replace each word ak,, which has length l k j , by a word akJ of
length ik, as follows. If a k j = u " J + ~ for z~ E {ul,. . . , u k i ) , then ük, = U ' ~ J + ~ :
the same procediire applies if ak, is a power of t E { t !, . . . , tk, }. 'iow let ak,
be a maximal shrinkable word of the third type. meaning that a subword of
a k j is close to the u-side q of a middle t-strip. Let P be the initial point
of the path corresponding to the word ak, and Q its endpoint. Denote by
Pt and Q' the phase vertices closest to the projections of P and Q ont0 q .
respectively; the path P'Q' represents the subword U ' ~ J + ~ of q. Replace an
arbitrary entry of u'v in P'Q' by U ' I J . making the appropriate changes to the
diagram PP'Q'Q. Then àk, is the word corresponding to the path PQ after
t hese changes.
Replacing al1 the words a,k by EIk, we again get a solution of the original
systern of equations (see 123, Lemma 111).
When we cut a power of u out of a long pseudoangle of a middle, we also
have to cut the corresponding cells out of the middle. Here we should make
sure that no matter where on the path EF (Fig. 1'7) we cut out un, the effect
on the word BC will be the sarne . More precisely,
Lemma 3.9 In the diagram s h o m in Fig. 17 a), €S-lcb2= b l ~ 1 6 1 L (in
other words, after removzng the first copy of un, w e get the same result as
after remouing the second copy) .
Figure 17
Proof Indeed,
6, < * c ~ = = ';
Multiplying (3) by E and replacing EU"+' in the right-hand side by the
left-hand side of (2), we obtain the required equality. 0
The proof of the following lemma is now the same as that of [25. Propo-
sition 11.
Lemma 3.10 It is possible to d e t e m i n e a recursi-ue function $(m) such that
if a system of rn triangulur equations has a solution in GQ , then it also has
a solution in a group H+(,), where Ho = G and is obtained from H, b p
adding a root.
The lemma implies that we c m take = w ( r n ) . By a reduction
procedure identical to that of [25, Section 61: we can effectively reduce
the original system S to a finite set F of systems in the hyperbolic group
G = G t ( t l , . . . , tu(,,), with the restriction that sorne of the variables belong
to subgroups generated by only a part of the generating system of G (cor-
responding to the requirement that u ~ , + L , . . . < uk,,, € H i ) . This restriction.
however. does not prevent us frorn applying the Rips-Sela method to test
each system from the set F For solvability. Indeed, if a system has a solution
.Yl,. . . ,Sr. with some S, belonging to the group G * (ti.. . . . t,). then the
Rips-Sela representative of -Y, is forced to be in the group F * ( t l , . . . , t , ) .
(Here F stands for the free group with the same generating system as G.)
Therefore. in the Rips-Sela method, the system over the hyperbolic group
with the restrictions described above gets reduced to a system over a free
group with the same restrictions, which is algorithmically decidable by (431.
This completes the proof of the Main Theorem.
4 Lyndon's group is conjugately residually free
Lyndon's group ~ ~ [ ~ i is the free exponential group over the ring of integral
polynomials Z [ x ] . This group, introduced by R. Lyndon in the 1960s. con-
tinues to be of interest to algebraists due to its importance in the stucly of
first-order properties of free groups, in particular, equations over free groups.
One of the crucial results of Lpdon ' s study was that the group is fully
residually F; i.e., for finitely many nontrivial elements in ~ ' ( ~ 1 there exists
a homomorphism p : F F ~ ] -t F. which is the identity on F, such that al1
the images of the given elements under p are nontrivial. 'vforeover. it was
recently proved by O. Kharlampovich and A.XIyasnikov ('27) that a finitely
generated group is fully residually free if and only if it is embeddable into
F ~ [ Z I . In this chapter we study residual properties of F ~ [ ' ! with respect to
conjugation and prove tha t it is possible to map ~ ~ l ~ l to F preserving the
nonconjugacy of two elements.
4.1 Preliminaries
Here we collect some essential definitions and facts which will be useful in
the rest of this chûpter. Recall that the definition of an esponential group
and the tensor completion G:' of a group G over an associative ring A with
identity can be found in Chapter 2.
Definition 4.1 L yndon's group F ~ [ I ] is the tensor completion of a f ~ e grovp
F over the ring of polynomials in one vafiable wzth integer c o e ~ c i e n t s , Z [ x ] .
In [XI] A. Myasnikov and V. Remeslennikov gave a detailed description
of the structure of the tensor completion of a CSA-group G over an arbi-
t r a ~ ring il of characteristic O. Their construction applies. in particular. to
~ ~ [ ~ i m d involves HNN-extensions of specific type, callecl free extensions of
centralizirs.
Definition 4.2 Given a group G and an elernent II E G with ubelian cen-
tralizer C ( u ) , the jree extension of the centralizer of u is
G(u, t ) = (G. t 1 t%t = U! u E C(ü))
Some important results of the next section are based on the explicit con-
struction of F ~ I ' I , thus it is justified to describe it here.
Let Gi = F be a finitely generated free group. For i 2 1. let =
{ai,, a,, , . . . , ain } be a maximal set of primitive reduced elernents of Gi whicli
have painvise noticonjugate centralizers and length less than i+l in G, (diere
the length of a word w is the nurnber of letters in the reducecl form of c).
CVe now form Gi+t from Gi and I..;+I by extending the centralizers of each
element in I-iiL. Le.
Then F'['] = UI,, Gi. It follows from the construction (see [SOI) that
~ ~ t ~ l is a CSA-group, and hence al1 of its centralizers are maximal abelian
malnormal subgroups.
In the proofs later on, sometimes it will be convenient to write elements of
an extensions of centralizer G(a, t ) = (G, t 1 t-% = U , IJ E C(u)) in normal
f o m or in power nonnal f o n . Let us clarify what w e mean by either term.
Definition 4.3 The normal form of an element w E G(u, t ) is simply the
normal f o n ofw as an element of HNN-extension, which ezists and it unique
by [do/. More preczsely, let us fix a set of right coset representutives of C ( u ) in
G. An element w E G(u, t ) is in normal f o m ! if iu = gotQ - - P g , ( n 2 0 ) .
where the sequence go, tCL, gl, . . . , t cn , gn zs such that
(1) 90 E G,
(2) for i = 1 , . . . , n gi is a n'ght eoset representative ofC(u) in G ,
(3) there 2s no consecutive subsequence t t ' , 1. t-'.
If in normal form w = gotq . tcng,, then we define the length of rx to be
Jw( = n.
Definition 4.4 (Power Normal Form) Let 1 ~ . E G(u,t). Thcn I.C is in
power nonnal f o n , if w = p c ~ c ~ . -cT where:
(1) P E C d 4
(2) cl, . . . , c, are either powers of t or riglrt coset representatives (us chosen
before for the normal form) and Qi, q # 1
(3) Qi < r , i f ci zs a power of t , then q + l is a right coset representative, and
vice uersa.
It is clear that the
the normal form of W .
power normal form of
We now define length
74
w can be found uniquely from
for a power normal forrn.
Definition 4.5 Let u: E G(u, t ) be vni t ten in power normal f o m : u.1 =
pclcz **-*. c,. T h e n the power normal length of w , lw l p = r .
Definition 4.6 Let u = uOtQ1ul ta? . . . tanun and u = ~ p ~ t ~ ~ ul td? . . . L*,- l t J m C,
be two elernents of a free extension of centralizer G* = (G. t 1 [C'(a), t ] = 1).
wn'tten in power normal f o n . Consider the product uau. CVe sag that a
reduction occurs be tveen u and ,v in w u , i f for sorne j 2 1 either
with w E G \ C(a ) , o r UV = uo ta i~u l . . . tan- ) w with ,u: E G ( j = m ) , or
~ û ' = w ~ ' J + ' , u ~ + ~ . . . tiimu, with LL' E G ( j = n). In each cuse, we trrnl
undj tan-^+' . . . un- tan un (U~~'L,U~ . . . t4 uj) the part of u (respectively. C ) in-
oolved in the reduction.
Next ive give a theorem on conjugacy in a free extension of centralizer.
which is based on lemma 2 in 3Iyasnikov and Remeslennikov [SOI.
Theorem 4.1 Let g be a qclicallg reduced element of
G(u . t ) = (G, t 1 P r t = 2 . 2 E C(u)).
(1) If g i s conjugate t o a n element h in G, then g i s also in G and is conjugate
to h in G.
(2) If g = ggottLgl + ---Yn, n 2 1 and h zs a conjugate cyclically redvced element.
then 191 = 1 hl und g is a conjugate of sorne appropriate cyclic permutation
h' of h by sonle element c E C(u).
We will also need the following theorem on conjugacy in a free group,
found in Lyndon and Schupp [do].
Theorem 4.2 Let F be a jree group with generators xi . . . ? x,. Two u~ords
CVl, Li; E F are conjugate if and only il thear cyclic reductions are cyclzc
pennutatzons of each other.
4.2 Results
We now introduce the concept of being conjugately residually free.
Definition 4.7 Giuen two groups G' and G, we say that G' is conjugately
residzlally G 21 for an3 nonconjugate g, h E G* there erists a hornomorphism
4 : G' -t G svch that d(g) und Q(h) are not conjugate in G.
Theorem 4.3 (Main theorem) Lyndon's group F'&/ is conjugutelg resid-
va& free.
Let F be a free group and H the group obtained from F using a finite
sequence of free estensions of centralizers:
F = G o GI = (Go. tolt;':to = 2 VZ E C ( u o ) ) 5 . . .
5 G, = ( G , - l , t n - l l t ~ ~ l A , - l = 2 Qz E C'(a,-1))
G,+i = (G,! t,lt,'zt, = z V: E C(a,)) = H.
For any two elements g and h of F ~ I I ~ , there exists a subgroup H of F ~ ! ~ I
which is obtained from the free group F by a finite number of free extensions
of centralizers and contains both g and h. If g and h are not conjugate in
F ~ [ ~ I , they are clearly not conjugate in H; moreover, any homomorphism
I) : H -t F with the property that $(g) and $(h) are not conjugate in
76
F can be extended to a homornorphism # : F'&I -t F thnt preserves the
noncongujacy of g and h. (Indeed, if .W E F'[=I \ H, w belongs to Gjcl =
(Gj, t j 1 [C'(aj), t j ] = 1) for some j > n; we let 4 ( w ) = y? O en+, O . . . O ~ ( u ) .
where for i = n + 1.. . . , j Qi : Gi+L -t Gi is the homomorphism sending the
elements of G, tù tlieiiiselves üiid l , tu ini arbitrary pover of ai. j
It is therefore sufficient to prove the following theorem:
Theorem 4.4 Any group H obtuined h o r n a free group F bby a finzte sequence
of free extensions of centralirers zs conjugatelg residuufly free.
O ~ c e the group H is fixed, we may assume, rearranging the order of ele-
ments a*, . . . , a, if necessary, that theSchain of groups F = Go < GL 5 . . . 5
G,, = H satisfies the following property: ai , . . . , ski E GO: uk, + 1, . - . . L L ~ ? E
Gl \ Go, etc., so that for any given s, we do not deal with the centralizen
of elements containing t , until we have extended the centralizers of those
that contain only t, for r < S. This implies, in particular, that uhen s e
estend a ce~itralizer C that is not. cyclic, i.e., C = (a. t,, . . . . , ti, ) and none
of t i l t . . . . ti, occur in a . then C is cienoted by C(a) in the coristruction. Ke
may assume, in addition, that in the notation above the indices il!. . . , i, are
in direct succession to each other, Le. i l = , + 1. Finally, for each i me
may consider a b i f l , . . . , ski+, to be painvise nonconjugate, cyclically minimal,
and primitive (i.e., not proper powers). Here, for the sake of consistency of
notation, we put ko = 0.
We are now ready to prove Theorem 4.4.
F k two arbitrary elements g and h of H such that g and h are not
conjugate. We may assume that both g and h are cyclically reduced, as g
and h are conjugate if and only if their cyclic reductions are conjugate. PVe
will proceed by induction on n, i.e., assume that the groups Gi for i < n+l are
conjugately residually F and construct a homomorphism $J : H = Ga+, -t Gi
for some i c n+I such that the images of g and h under iC) are not conjugate.
The desired homomorphisrn 9 : H -i F niIl then De the composition ùf G
and a homomorphism ~' : Gi + F preserving the nonconjugacy of @(g) and
#(h ) , which esists by the induction hypothesis.
To obtain the basis for the induction, we first need to show that eveqi
group obtained frorn F by a single free extension of centralizer is conjugately
residually F. This case was considered in detail in [7]; we will provide the
proof from [ï] liere in order to make Our discussion self-contained.
Theorem 4.5 Let F be the free group with generators {x i , . . . , x,}. T h e n
ony group of the / o n F(a . t ) = ( F , t ( [ c , t ] = 1. v E C(a)) is conjuptely
residuall y F.
Proof Take any u , a E F ( a , t) sucli that utc are not conjugate. We nniay
assume that both u and c are cyclically reduced. There are three cases to
consider. In the following Ive will write IL. u in power normal form.
In the first case u = ggtklgL t k n ; u = hotLlhl * - d l r n where n # m.
We will choose our homomorphism # to be such that the generators of the
free group are mapped to themselves and 9(t) = a', where z is some large
integer. Notice that gi, . . . , g,, hl , . . . , hm C'(a), as we are in power normal
form, and that if go E C(a) (ho E C(a)) , then we must have that n = 1
(m = l), othenvise lugo'*' 1, < Iulp (sirnilady I V ~ ~ ~ ~ 1, < Ivlp). Thus if we
write gi, h, so that al1 powers of a at either end have been separated out we
have gi = a" gjasi , hi = axa hi au' and t hus
wherc., if z is chosen largo enough, no power of a is trivial (except maybe
aro, aZ0, which have no effect on the argument.)
CVe now examine the cyclic reductions and see that if ro is the opposite
sign from k,z + s,-i, the cyclic reduction of #(IL) is
otherwise u was already cyclically reduced. Similady for v, if the signs of xo
and I,r + y,- are the same, then d ( u ) is already cyclically reduced. otherwise
the cyclic reduction is
Xotice that if z is large enough. none of the poeers of a in the cyclic reduc-
tions is trivial, hence the two cyclic reductions of @(u). d(u) are not cyclic
permutations of one another, as one has more occurrences of powers of a than
the other. Hence from Theorem 4.2 we have thnt O ( u ) is not a conjugate of
4w* In the second case u and v have the same lengths, but the powen of t
in u and o are different no matter which cyclic permutations of the elements
are taken. We will write u and v as above, only now consider that n = m.
Thus looking a t the cyclic reductions we have (depending on the sign of Q)
that O(#(u)) = a ~ g @ i ~ + s ~ + r i k~=+s l+r2 glu
1 a k n ~ + ~ n - l 'Sn-i
and similarly for u. Clearly, if z is very large, the 4 2 ( l l z ) terms dominate
in the exponents of a, and thus by picking a large enough value of z we can
ensure that any two powers of a in <r(@(u)) and a ( d ( u ) ) are different if the
coefficients k, and I i are different. Hence, looking at each cyclic permutation
.u' of u , we can choose a value of t which ensures that the powers of a in
a(6(u8)) do not match the powers of a in a(# (u ) ) . Picking the largest such
value of z , we find that the powers of a do not match in a(&u)) and a(~(u))
for any cyclic permutation of u and v. As before, we have from Theorem 4.1
that d(u) is not conjugate to @ ( u ) in F.
The third case is when the elements u and *u have the same lengtii and the
same powers of t for some cyclic permutation (we will write them so that we
use one of these permutations). Orice again we will map t to a large power
of a
$(u) = goa'Lgla'2 - . *gn-la'n
where = kiz .
If these two elements are conjugate, then we must have a c E F such that
We can now consider two cases, either c E C(a) or c C(a)
If c E C(a) , notice that in order for the aZn-'-terms to be cancelled we
must have that gn- laznca- 'nh~~L E C(a), i.e. as c E C(a),
where ci E C(a). Further, to cancel the terms me must have that
However, it is easy to show (see, for example, Theorem 3.3 in [il) that if
these equations are solvable, the original u and v were in fact conjugate.
Thus if the two elernents are conjugated by c we must have that c @ C'(a).
We will assume that ho1 is not cancelled by c-' (if this is not the case, a
simiiar argument to the one below applies.) CVe consider what happens if
the term c cancels al1 or part of the term 9 ( u ) before it and al1 or part of
4 ( ~ ) which occurs after the c-term. We will write $ ( u ) = ~ 1 2 ~ 2 ~ Q(U) = v p ?
and assume c = UT' c2. Then, by Theorem 4.2, the two conjugate cyclically
reduced elements in the free group are cyclic permutations of one another.
simplifies t o
-1 -1 U l V , v, u2 = 1
and t hus
(where ul, u2, V I , v2 can be 1). There are now three cases to consider: in
case 1 both ul;ul end in a power of a; in case 2 both u1;ul end in a coset
coefficient; and in case 3 either ul = go or .vl = ho while the other ends in a
power of a. Notice that in the following powers of a must occur in the same
order in both cyclic permutations of the eiements due to Theorem 4.2 (i.e..
although it is possible that r # s, we do have that =aL*+& V i , where we
consider a" = a Z ~ + ( n - ~ ) + l = aZs+(n-*)+l). In the first case Ive have
while in the second
In both cases, in order for this to be true, the following system of equalities
must be true where a, E C ( u ) :
However, this implies (see, for example, the proof of Theorem 3.3 in [ï]) that
the elements = g,t=r+L . . . 9,- lt=" (Jottl . . . gr- t"
and
are conjugate; it follows that u and v are conjugate as cyclic permutations
of the elements are conjugate, hence in the above two cases 4(u) and $ ( u )
cannot be conjugate.
In case 3 ive will look at the case when ul = go; the case ,cl = ha is
similar. Once again the equality of powers of a follows as above. Then we
have the equation
We now arrive a t the following system of equations, where ai E C(a)
This means that h, = ho, as only ho may belong to C(a ) (it also implies
that both u, v have power length 1). But ive now have that in fact c E C(a),
a contradiction. Hence b(u), 4(v) are not conjugate, and thus F(a, t ) is
conj ugately residually F. CI
It is easy to see that the argument above can be slightly rnodified to
show that any group Gc+i obtained from F using finitely many extensions
of centralizen of elements of F is conjugately residually F; more precisely,
Grti is constnicted as follows:
where al1 ai for i = 1,. . . , k are elements of F. We wi!l use this, more general.
statement as the basis for Our induction.
For the rest of the proof, Ive will assume that H = Gn+i is constructed as
in the beginning of this section, and a, $ F. For any two elements g. h E H
that are not conjugate in Hl Ive will construct a homomorphism @ : H + G,
for some 1 < n + 1 such that $ ( g ) and ~ ( h ) are not conjugate in Gi. In what
follows, unless stated otherwise? g and h will be mritten in power normal
form.
We have three cases to consider:
Case 1 g = got;lgi.. . tnmg,, h = hotnlhl . . . t? hk where k # m.
Subcase 1.1 The element t,-i occurs in a,, Le., a, E G, \ G,-l.
In this case the approximating homornorphism w maps G,+l to G,. CVe
define $J to be the homomorphism G,+l + G, induced by the mapping that
sends the elements of G, into themselves and t, into a i for some large integer
z. Let us show that if r is large enough, +(g) and $(h) are not conjugate in
G,. For simplicity, we will denote t, by t and a, by a.
Taking cyclic permutations if necessary, we assume t hat g = gOtaLgl . . . tam
and h = hot@lhl . . . tek, rn # k. Notice that gl, . . . , g,+ hl , . . . , h k - ~ $Z C(a )
since we are in power normal form; moreover. Yi and hi for i > O are non-
trivial right coset representatives of C(a), therefore we c m write gi = g ~ a r ~ .
hi = hias#, where g:, hi $ C(a ) and do not start with a power of a. Observe
also that if go E C(a) (ho E C(a)) , we must have that rn = 1 (k = 1):
othenviçe [ggotaL l p < iglp (similârly, ~h~"'' ' l p < !hlp). Writing and h, so
thst al1 powers of a at either end are separated out, we have go = argbar*.
Consider now the images of g and h under i :
where, if r is chosen large enough, al1 the esponents :ai + rial. i = 1.. . . . m.
and zgj + sj -1 , j = 1. . . . , k, are non-zero.
It readily follows that the cyclic reduction o(@(g) ) of ,d'(y) has the form
g;a"gCaz? . . . g z - ,azm. where X I ? . . . , z,- 1 differ from rai+ ro. . . . . -a,- 1 + rm-2 (and x, from ;a, +r,-~ + r ) by at rnost 1; similarly, the cyclic reduc-
tion o(@(h)) of @(h) has the form h:aYt hlaY2 . . . hg-,aYk, where 91: . . . . yk- 1
differ from zPi + so, . . . , + sk-2 (and yk frorn z& + sl;-l+ s ) by at most
1. If z is large enough, none of the powers of a in the cyclic reductions are
trivial. Since the number of occurrences of powers of a in the cyclic reduction
of +(g) is different from that in the cyclic reduction of #(h) , o(,@(g)) can not
be conjugate to a cyclic permutation of o($(h)) by an element of C(a,-l).
Therefore, by theorem 4.1, @(g) is not a conjugate of $(h).
Subcase 1.2 Let j < .n - 1 be the largest number such that t j occurs in
a,. We will construct a homomorphism : G,+l + such that +(g) and
# ( h ) are not conjugate in G,+i. For every s E {j + 1,. . . , n) denote by t,bz,
the homomorphism from Gy+i into G, induced by the mapping that sends
eiements of G, to themseives and t , to aia for some large number 2, E 2.
Denote by li, the composition of these homomorphisrns: ri, = +J,l o . . . O L!,.
By [5] G,, is y-residually G, for every s E { j + 1, . . . . n - 1} : more-
owr, the discriminating hornomorphisms from G,+[ to G, are exactly of
the type .&,. Since in G, we have [gola,] # 1,. . . , [g,-ilan] # 1, [ho,a,] + 1, . . . , [hk- l , a,] # 1, it follows that for an appropriate choice of Zn-, the same
finite set of inequalities will be valid in G,-l for the images of go, . . . .y,- l ' ho
. . .. hk-1 under I) ,,-,: [gr in-Lt~n] # 1,. ..! [g:2yi,an] # an] # *zn-,
1, . . . , [hkdL , a,] $ 1. Similarly, the rest of zi can be chosen in siich a way
that in Gj+l [y,,a,] # 1.. . . , [g,,+17a,] # i , [ L a n ] # 1, ... , [ z k - l d h ] # 1.
where gi (hi) stands for the image of gi ( h i ) under the composition of homo-
morphisms iItZJ+, o . . . o &,-,. The rest of the proof is now the same as for
Subcase 1.1: if 2, is large enough, we can show that the nurnber of occur-
rences of powers of a, in the cyclic reductions of $(g) and $(h) are different.
and thus $(g) and $ ( I I ) are not conjugate in Gj+1.
Case 2 g and h have the same lengths. but the powers of t, in g and h are
different no matter which cyclic permutations of the elements are taken.
We construct the approsimating homomorphism 1/, in the same way as in
Case 1, depending on the largest j such that t j occurs in a, = a. As before,
the cyclic reduction of Q ( g ) has the fonn
and similarly for cr(@(h)). (Here z stands for Zn if the largest t in a is not
t . For large z. the terms -cri ( ~ 0 ~ ) dominate in the exponents of a. and
thus by choosing a large enough value of z we can ensure that any two powers
of a in a($(g)) and a(@(h)) are different if the coefficients ai, are different.
Taking the maximum value of such z over al1 cyclic permutations of g and
h, we find that powers of a in o(+(g) ) and cr($(h)) do not match no matter
which cyclic permutations of the elements are taken. Again? by theorem 4.1!
this irnplies that $(g) is not a conjugate of @(h).
Case 3 g and h have the same length and matching powers of t, for some
cyclic permutation; making use of that permutation. we can write g and h
as follows:
Subcase 3.1.
Here we assume that a, is an element of G, but not G,-l, i.e., t,-, occurs
in a,; it follows from the choice of the elements ao, . . . , a, that [a,, t,-l] # 1.
In this event, C(an), the centralizer of a, in Gn , is the cyclic subgroup of
G, generated by a, (a, is not a proper power by construction).
By the assumption of the theorem, g and h are not conjugate in H =
Gn+l. We will construct a homomorphism $ : G,+i + G, such that @ ( g ) and
$(h) are not conjugate in G,. Let $ send the elements of G, to themselves
and t, to a:, where z is a sufficiently large integer; it will be determined
in the course of the proof exactly how large z should be. It is clear that
this mapping induces a homomorphism 11 : G,+i -+ G,. We will sometimes
denote tliis Iioniornorpliism by 3: to emphasize the fact that its definition
depends on z.
&(g) = goangl ..... aimg,,
Qr(h) = boa: h l . . . aim hm
(ti = z<ui.)
Suppose that for every z &(g) and S(:(h) are conji
exists f E Gn such that
lgate in G,, i.e., ther
f - ' goangl . . . g , - l ~ Z g , f = h o a 2 h l . . . hm-~n~mh,.
Consider a Van Kampen diagram over G, corresponding to this identity
(Figure 1.) For simplicity, Ive vevill now denote a, by a and t,-l by t .
In what follows, we need to be able to increaçe indefinitely the t-length
of a' by increasing z, but in such a way that the t-length 1 f 1 of f remains
bounded. In general, Iiowever, f depends on z , and its t-length may grow to-
gether with z. To get around this difficulty, we will make use of the following
lemma:
Lemma 4.1 Assume that for e v e y hornomorphism @: : G,,, + G, ( t ~ )
and $,(h) are conjugate in G,. Then /or any positive integer ;Y there ex-
ist a number z, an element f in G, and a cyclic permutation v ~ ( g ) = -
1 goaz;g; . . . 9,- ,azLg& of @, (g) such that f ( t ~ ) f = *t,bZ ( h ) in G,, la': 1 > :V
for al1 i = 1,. . . , rn, and in the vord f - l g & f ; no reductions oecw betzueen
f -' and a';.
Proof Consider a homomorphism li, = we, where z is large enough so
that al1 a+ for i = 1, . . . , m have length greater than A'. By assumption.
there esists f E G, such that f - 1q9 (g ) f = *&(h). Let us write f and $(g )
in power normal form in ~ h e HNN-extension G,: f = f&'lAt@?f2 . . . twr f,.
zj(g) = datsldl td? . . . tLd,. If no reductions occur between f and @ ( g ) ,
there is nothing to prove; otherwise let <l(g) = d ~ ( g ) 1 tb(g)*< where .@(g) 1 is
the part of $ ( g ) involved in the reduction with f -'. Assume for simplicity
that ends in a power of t (the same argument directly translates to
the case where @ ( g ) , ends in an element of G,-l), so we can write + ( g ) 1 as
dotdi d i t A . . . djWlt' . Similarly, f = f' f ", wliere (f ')-' is the part involved in
the reduction, so, according to the assumption above, f ' = fot4' fi . . . fjt4j .
For a reduction to occur, we must have: fado = ua E C(a , - l ) ; 4' = 6'. Then
-4s before, f ; ' u0d l = E = d2, and so on. Proceeding in the
same fashiono we arrive at the following expression:
and, finally, I,%,-ld, = iLj E C(U,-~). dl = d,, so we have f - ' , ~ ( ~ ) f =
( f " ) - L ~ ~ j ~ ( g ) 2 f . NOW, if For some k the beginning of the subword an of $ ( g )
belongs to $ ( g ) while the end (denote it by a") belongs to i ( g ) 2 , take
( U " ) - ~ I L ~ ~ to be our new f , othenvise just take f = u 7 ' f ". The word $(g) is -
then replaced by its cyclic shift $(g) = gkaZk+l . . . ~ ' ~ g , g ~ a ~ ~ . . . a'*, or. if gk
was broken apart by the reduction (gk = gk.;), $ ( g ) = . . . ~ ' ~ g , ( l o a ' ~
. . . aZ*gk. -
In the new notation. f - l @ ( g ) f = y(h) , a':, which are simply a'. written in
a different order, satisfy the length requirement, and there are no reductions
between f and a'; = azC+i. [7
Lemma 4.2 Let Gj+' = (Gj, t j 1 [C'(aj), t j ] = l), j 2 1, and a E Gjci \ G,
a cyclzcally reduced primitive element. Wr i t e a in normal f o n in the group
Gj+i : a = botC1 bl . . . bs-l tes b J , C i = A l . Let A be a Van Kampen diagram
over Gj+L corresponding t o a n arbitrary equality of the f o m S Y Z = W wzth
the property that the path corresponding to kV in A has a subpath q &th
label afl for some integer P 2 2), and the path corresponding to Y has a
subpath p vzth label aa fial 2 IpI; see Fig. 2.) Suppose further that for euery
occurrence of t f l in a@ there zs a t j -s t r ip that connects t," with an occurrence
of t:' in aQ. Then for every subpath q l O/ q with label a there exists a subpath
pl of p with label a such that q, and pl are connected b y tj-stnps in rnatching
spots, i .e . , for every i = 1, . . . , s there zs a t,-strip Si that connects tj' in ql
Proof For simplicity, we twill denote t j by t . Taking a cyclic permutation
if necessas, we ni- assume that a starts with the letter t, so Ive can write
a = tcL b l t t2 b2 t c 3 b 3 . . . tcs bl. Suppose, contrary to the statement of the lemma.
that the top and the bottom a-paths are shifted with respect to each other
by i "units" . More precisely, let pl be the fiat subpath of q with label a and
suppose that the first entry of t in ql is attached by a t-strip to the (i + 1)-st
entry of t in a subpath pl of p with label a. Observe that in this case t'$-s+L
in ql is connected to teL in the ne.* subpath of p with label a; denote this
subpath of p by p? (see Figure 3.)
Note that since every t-ce11 in the diagram is labellecl by a word of the
type t-'wt = w with w E C'(aj) , the tvvo side labels of every ?-strip are the
same. Let q2 denote the second subpath of p with label a (it exists since
!O! 2 2.) For each i = 1.. . . . S . denote by î l i the side label of the t-strip
originating at tCL in ql and by u: the side label of the corresponding t-strip
for q?. (See Figure 3.) We da im that ul = u;.
Figure 3
Indeed, from the diagram in Figure 3 we clin read the following identities:
where al1 indices are taken rnodulo s, escept that u, and b, keep their notation
instead of being renamed into u o and bo.
To prove the claim, we use the following algorithm. Denote by d = ( S . i)
the greatest common divisor of s and i. Csing the first equation. express bl
in terms of b i+1:
bl = U ~ ~ ~ + ~ U ; ' .
Then use the equation in nhich bi+l appears in the right-hand side (namely,
uii1 b2i+ = bi+ 1 ui+.>) to express bii 1 in terms of b2i+l and substitute the reçu1 t
in the first equation:
bl = u ~ u ~ + ~ ~ ~ ~ + ~ u ~ ~ u ; ~ ~
Proceeding in the same fashion, we express each bki+l in terms of b(k+l)i+l and
substitute the resulting expression in the equation obtained at the previous
step, until at some point b ~ k + l ) i + l = bl. It is easy to see that this happens
when le + 1 = 2. Thereby, we obtain the following equation:
or, equivalently,
where indices are taken modulo s and k = 3 - 1. For our purposes. it will be
more con~enient to write this equation in a slightly different form. It is easy
to notice that the hl's that participated in this process are esactly b l . b l + d .
. . . , bl+bd. Each bl contributes u( to the right-hand side of the equation and
ulil to the left-hand side (an exception is b,, which would contribute ,u', to the
Ieft-hand side; but 6, does not appear a t the first step of the procedure unless
d = 1.) Moreover. the order in which we write the u j ' s does not matter. since
they a11 belong to the same abelian subgroup C ( a j ) of G,. Therefore, we can
rewrite equation (4) as
.Ci U2Ud+2U2d+?. . . U>t~+2 = b ; L ~ l ~ d r l ~ 2 d + I . -. u ~ ~ + L ~ ~ ? k = - - 1-
d
Honever. uq . . . ukdi- and u i . . . ukd+l both belong to C'(a,). Since G, is a
CSA-group (as a subgroup of Lyndon's group). the fact that C ( U , ) ~ L n C'(a,)
implies bl E C(a,). Thcrefore, bl cornmutes with each ul, and hence
Now for each 1 = 2 , . . . , d we repeat the process: express bi in terms of
bi+l . then bi+f in terms of b2i+I, etc. In a similar way as before, this gives us
the following system of identities:
This immediately implies >ul = u;, as claimed.
Denote by u the word b;'t-', . . . b ~ ! t ' + L u ~ l . By the ûssumption of the
lemma and the claim above, w-'a,w = a. Since the centralizer of a is a
malnormal subgroup of Gjil , it follows that [w, a] = 1. Moreover. Cc,,, (a)
is cyclic generated by a, so w = ak, but the t-length of w is less than thüt
of a - a contradiction. Thus, the only remaining possibility is that a =
t'b+l bi+1 . . . tCsb, , Le., i = 0. and the top and bottom a-paths in the diagram
are indeed aligned with respect to each other. Cl
We now continue the proof of the theorem for Subcase 3.1. Let us f i l
arbitrarily an infinite set of integers C from which we will choose the values
of 2 to construct a hornomorphism & with desired properties. For every
r E X fis an element ft such that qL(g) and +Jh) are conjugate by f,.
Suppose first that there exists a positive integer .V such that for every 2 E X
1 fJ 5 N. Note that g, and h, are fixed elements of G, and thus have bounded
lengths. Choose 2 E X large enough so that
and denote the corresponding f, by f . Considering the Van Kampen diagram
over G, in Fig.1, corresponding to the identity
we observe that the number of t-strips originating from f is bounded by N,
and the number of t-strips originating from go, gi, ho, and hl are bounded
by lgO1, IgL 1, Ihol, and lhl 1 respectively Due to the choice of z, there exists a
subpath q of a" on the bottom of the diagram with label aY1 and a subpath
p of azl on top of the diagram with label aZL such that 1 ~ ~ 1 ~ lyil 2 2: and
every occurrence of t k L in q is connected by a t-strip to an occurrence of t21
in p. Therefore, by Lemma 4.2, the a-paths p and q are aligned with respect
to each other, i.e., the diagram h a the form illustrated in Figure 4.
Figure 4
If there is no number M such that for every 2 E X 1 f , 1 5 'i, we do
the following. Choose r E C to be large enough so that la''] = la'QbI >
ma~{lgi -~l + lgil, Ihimil i lhiI) + 21~1 for al1 i = 1,. . . , m; denote the corre-
sponding f, by f . Using Lemma 4.1, Ive can replace f with another element 1 -' 1 of G, and (9) by its cyclic permutation @.,(g) = . . . g,- ,a-mg, in
such a way that f -l+= (9) f = & (h) , a:; is sufficiently long, and there are
no reductions between f - L and a'; - meaning that in the diagram in Fig.1
there are no t-strips connecting f with the subpath of &(g) corresponding to
a';. Similarly, by further altering f , if necessary, and taking an appropriate
cyclic permutation $, (h) = haa'yh: . . . h ~ - , a Z ~ hm of @, ( h ) , we c m assure
that in the product ~ = ( h ) - ~ / - ' no reductions occur between a - 5 and f - l .
and thcrcforc in thc diagram in Fig. 1 thcrc arc no t-strips connccting f
with a<. As above, due to the choice of r , there exists a subpath q of a::
with label au1 and a subpath p of a'; with label axl such that lxl l ! lgi 1 2 2.
and every occurrence of t*' in q is connected by a t-strip to an occurrence
of t k L in p; again, we apply Lernrna 4.2 to show tliat the a-paths p and q are
aligned with respect to each other. Renaming gi as g, and hi as hl for each
i = 1.. . . , m, we can write equation (5) for this case in the form
Clairn 4.1 Let t and f = f, be chosen as in either of the two cases above.
Then [hil /-Lgo,a] = 1.
Proof As before, let us assume that a starts with a power of t . i.e.. a =
P b l . . . b ,JeS b,, bi E G,-L (written in normal forml so ci = f 1.)
Since [a, t ] # 1, there exists an i < s sudi that bi $ 1. Choose the
minimal i with this property. Consider the first two occurrences of a, one on
top and one on the bottom of the diagram, that are connected by a t-strip.
Let u be the side label of the t-strip connecting te' on the bottom with tek
on top, and v the side label of the t-strip that goes between tC1i-1 and PL.
Then we have a subdiagram whose boundary label reads b ; L ~ b i = II, mhere
u and v belong to the same maximal abelian subgroup C(a,,-') of G,. Thus
bi E but this contradicts the choice of b i s Therefore, u = 1 = u'
which implies that the side labels of the t-strips originating at t j for j c i
also vanish, and the diagram takes on the appearance illustrated in Figure 5
(contains a subdiagram with boundary label hi'f - L g o a k ~ ~ k 2 . )
Figure 5
Thus the following identity holds in G,: h i 1 f -lg0 = a"?-1 , and hence
[ h i L f -'go, a] = 1. Here we denote by akt the subword of a'' (u ' ; ) on top of
the diagram that is connected by t-strips with f and go, and by a" the sub-
word of azL (a'!) on the bottom of the diagram that is connected by t-strips
with f and ho. @
In what follows we Nil1 consider equation (5) in the form
the proof for the case
Using Claim 4.1, we can now simplify equation (5), replacing f -'go in the
left-hand side by hoa:'l (where hl1 stands for k2 - kl) and thus obtaining:
or, after reduc tion?
Gsing the same argument as in Claim 4.1 with f - l replaced with a-"L.
we obtain that aMtgl = hlaA". (Note that the length of a."' is bounded by
N + lho 1 + (go 1, so we no longer have to take cyclic reductions.) Therefore.
Proceeding in the same fashion? Ive finally arrive at the identity
Going back to the original elements g, h E G,, i, we recall that
k km 9 = g0tnLg1 . . . t","-ig,& g,
h = h,,tk,lhl. .. t ~ - t h , - i t ~ h ,
By the argument above, hm = u " ' ~ ~ , f, nhere [a, t,] = 1; therefore.
furthemore, since h,dia*4fm = cihfm-L gm-1,
and so on. Finally, as hoaJbfl = f -'go, we obtain f - 'g f = h; i.e.. the original
elements g and h are conjugate in Gncl, contrary to our assumption.
Subcase 3.2 Suppose now that t,-l does not occur in a,: the centralizer of
a, is not necessarily cyciic.
In this case, it wiil be convenient to replace the chain of groups F =
(3) Go 5 GI 5 . . . 5 Gnil with another chain GO) 5 GY) 5 . . . 5 Gn+
where GO) = F7 G!:, = (G~"), t!')l ai")), t?] = 1) for al1 i = O . . . . . n. and
G!$ iç isornorphic to G,+i; moreover, the chain of groups G!" satisfieç the
folloming condition (C): either a$) contains t i ! : l l . or j, is the largeçt nurnber
such that t j , occurs in a n ) , and for every i E { j , + 1,. . . : n - l} j, is the
largest nurnber such that t j 3 occun in aiy). To construct the chain of groups
G!"). we wili use the foiloming algorithm.
Step 1. CVe know that the group G,,, is obtained from by a sequence
of two extensions of centralizers: one extending the centralizer of a,-l and
the other the centralizer of a,. Sirice a, does not contain t,,,Lt both a,-[
and a, are elernents of G,-!. and hence the group G,,i is isomorphic to
cYil = (G,, t ' , ') i[C'(aY),t1)] = 1)' where ai1) = a,+ t : ) , and G, =
(G,-L, t , 1 [(?(a,)? t,] = 1). (Informally speaking. ive have changed the or-
der of the last two extensions of centralizers in the construction of Ga+[.
interchanging a, with an-1 and t , with t,-l .)
If n - 2 # j, we use the same argument as above to "interchange" a,
with an-2, now taking G, instead of Gncl and Gn-* instead of G,-,. More
generally, we repeat the procedure for every k such that 1 < k < n - j . If k
is in this range, a, does not contain t,,-t, therefore is isomorphic to
Step k + 1, k 2 1. Consider the element aLk) and let jk denote the largest
( k ) number such tliat t j , occurs in a(nk). If jk = n - 1 or for every
i E { j k + l? . . . , n - l} t jk occurs in ai and jk is the largest number with this
property, ive put s = k and declare the process complete. Othenvise. use the
05) same argument as in Step 1 to "interchange" a;) with a,+. . ., and. finally. ( k ) (k) ( k ) with aj,,,. More precisely, ive put GE:? = (GJk , in 1 [C'(a:'))! tkk)] = 1):
(k) (k) (k+l) = p Wl) ~ f + ' ) = Gu ; for i = O , . . ., jk - 1 aik+') = ai , ti S i + l =G,+i; (k)
(k+l) - (k) (k+L) - (k) and finally, for i = jk + 1'. . . , n a, - - 1 1 t i - t,- 1, and
Since the set {aa) , . . . , a n ) ) is the same up to permutation at every step
k (and finite), a t some step s the element a:) haç already occurred at a
previous step i; i.e., an) = a!) for some i E {O,. . . , s - I}, where an) stands
for the original a,. W e clairn that in this event the process terminates at step
Y, i.e., the chah of groups GO) 5 . . . 5 G::' satisfies the required property.
Indeed, if j, is the largest number with the property that t j , occurs in a!)
($1 and j, # n - 1, then, by construction, for every i = j, + 1,. . . , n - 1 ai
contains t j , and does not contain tk for k > j,. It is also clear from the (4 construction that GO) = Go and G,+' - Gnii.
From now on, we will assume that the chain of groups Go 5 G i < . . . 5
G,+, , where Gii1 = (Gi, ti 1 [C(ai), t i ] = 1), satisfies condition (C). AS before,
let us denote by j the largest number with the property that t j occurs in a,.
By condition (C), al1 ai, i = j + 1,. . . , n contain t j and do not contain Q
for k > j . Therefore, aj+i! . . . ? a, are elements of G,+ 1 \ G, arid we cun
represent G,+i as
IF g E G,+l, g c m be written in the following form:
where i l , . . . , i, E { j + I l . . . ? n} and gk E Gjcl for al1 k E {O.. . . , T B } .
bVe will s a y that g is in power normal form, if
(1) for al1 k E {l.. . . ,m}? O , # 0;
(2) for al1 k E (1' . . . , m - l}, if ik = it+1. then g i is a right coset represen-
tative of C'(ai,) in Gj+1 and gk # 1.
(3) for al1 E (1, . . . . m - 1). if ik # i k + i . then gk is a right coset representative
of C(ai,) in Gi+l (possibly trivial.)
Now let g and h be nonconjugate elements of G,+l. Let us write g and h
in power normal form as defined above:
h = hotCL hl . . . h&tph,,
g k , hk E Gj+l for k = O,.. . ,m; il ,..., i, E { j + 1,. ..,n). Here we may
assume that the porvers of t ik7s in g and h match for some cyclic permutation
(othenvise it follows, similarly to Case 2, that it is possible to map G,+i to
Gjcl preserving the nonconjugacy of g and h.)
We will construct a homomorphisrn zC, : Gnil -t Gj+1 such that ~ ( 9 ) and
@(h) are not conjugate in Gj+i. Let $ send t i into ai' for every i = j+ 1. . . . . n
and the elernents of Gj,l into themselves. Sometimes we will also denote u
by Qr, where 2 = . . . , Zn). to emphasize the fact that its definition
depends on f. We claim that if y+l,. . . , z, are sufficiently large, then .ut(g)
and #L(h) are not conjugate in Gjci-
As before, we observe that
+(g) = hoal:hl . . . a c h ,
(Y& =
Suppose that there exists j E G,+! such that j - l w ( g ) f = h. Denote t,
by t . We ccan clioose . . ,:, large enough for Lemma 4.2 and Claim 4.1
to be valid (if necessary. taking a cyclic permutation of u(g) and ri.(h) and
replacing f with an element of less t-length). This allows us to apply the
same argument as in Subcase 3.1 to obtain the equalities
ai,gmf = hm-
Going back to the original elements g and h in G,+i. we observe (again.
in the same way as in Subcase 3.1.) that the equalities above imply thst
g and h are conjugate by the same element f . This gives us the required
contradiction. and the proof is complete.
5 Conclusion
In this thesis, ive explored several different aspects of constructible groups.
The most important result of Chapter 2 is a description, in terms of free
constructions. of the tensor completion of a free nonabelian solvable group.
It turns out that this completion is, in general, not solvable. The results
of Chapter 3 present original geometric methods which have allowed us to
reduce the problem of solving systems of equations in GQ, the Q-completion
of a torsion-free hyperbolic group, to the analogous problem in a hyperbolic
group. Our results suggest that this technique can be applied not only to
equations, but also to inequalities. Therefore, the next logical step in this
direction vould be to establish the decidability of the universal thcory of a
torsion-free hyperbolic group G' and, more generally, of the universal theory
of its Q-completion GQ.
The residual properties of Lyndon's group ~ ~ [ ~ l establistied in Chapter 4
of this thesis lead to another important question connected with first-order
properties of constructible groups. It is known that every universal formula
with constants in a free g o u p F is true in F'[*] if and only if it is true in
F. However, the properties of the universal theory of F~['] with constants in
F ~ [ ' I are yet to be described. It appears that a combination of methods used
in Chapters 3 and 4 can prove to be a suitable tool for tackling the following
conjecture:
Conjecture 5.1 Consider a systern of equations in variables 3 wzth coefi-
eients in Lyndon's group: ~ ( I , E ) = 1, E E F ' [ ~ ] . Suppose that for e v e y
hornornorphism # : F ~ [ ~ ] + F the system f(2, @) has a solution in F. Let
G be a ~ubgroup of genemted b y the coeficients e. Then there exists an
ernbedding p : G + ~ ' 1 ~ 1 such that the system f(~? 3') = 1 has a solution in
~ ' [ 4 +
A proof of this conjecture would shed light on the question of "lifting
solutions" from F to ~ ~ [ ~ 1 and would be an important step towards the
understanding of first-order properties of groups close to free groups.
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